Surface Diffusion

Surface Diffusion Metals, Metal Atoms, and Clusters For the first time, this book unites the theory, experimental techniques, and computational tools used to describe the diffusion of atoms, molecules, and nanoparticles across metal surfaces. Starting with an outline of the formalism that describes diffusion on surfaces, the authors guide the reader through the principles of atomic movement, before moving on to describe diffusion under special circumstances, such as the presence of defects or foreign species. With an initial focus on the behavior of single entities on a surface, later chapters address the movement of clusters of atoms and the interactions between adatoms. While there is a special emphasis on experimental work, attention is paid to the increasingly valuable contributions theoretical work has made in this field. This book has wide interdisciplinary appeal and is ideal for researchers in solid state physics and chemistry, as well as materials science and engineering. Graz· yna Antczak is a Humboldt Fellow in the Solid State Physics Department at Leibniz University, Hannover, Germany. She received her Ph.D. from the Institute of Experimental Physics at the University of Wrocław, Poland, where she is now an adjunct researcher. Dr. Antczak is a Member of the American Physical Society and the American Vacuum Society, and has had 15 publications in scientific journals. Gert Ehrlich is currently Research Professor in the Department of Materials Science and Engineering at the University of Illinois, Urbana-Champaign. He is internationally recognized as a pioneer in the area of surface diffusion, and he has received numerous scientific honours and awards. Dr. Ehrlich is an active member of various societies, and is a Fellow of the American Physical Society and the New York Academy of Sciences. He has written almost 200 journal articles and has served on several editorial advisory boards.

Surface Diffusion Metals, Metal Atoms, and Clusters GRAZ˙ YNA ANTCZAK Leibniz Universita¨t Hannover, Germany

GERT EHRLICH University of Illinois, Urbana-Champaign

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sa˜ o Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521899833 © G. Antczak and G. Ehrlich 2010 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2010 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library Library of Congress Cataloging in Publication Data Antczak, Grażyna, 1973– Surface diffusion : metals, metal atoms, and clusters / Grażyna Antczak, Gert Ehrlich. p. cm. ISBN 978-0-521-89983-3 (hardback) 1. Diffusion. 2. Metals – Surfaces. 3. Surfaces (Physics) I. Ehrlich, Gert. II. Title. QC176.8.D5A58 2010 530.4′15–dc22 2009050502 ISBN 978-0-521-89983-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface List of abbreviations List of symbols 1

2

3

Atomic diffusion on surfaces

1

1.1 Diffusivities: an introduction 1.2 Distribution of atomic displacements 1.3 Jump rates

1 7 17

Determination of adatom movements

24

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

24 29 31 34 44 46 53 56

Field ion microscopy Field electron emission microscopy Scanning tunneling microscopy FIM measurement of diffusivity Displacement distributions STM measurements of diffusion Other measurement techniques Theoretical estimates

Atomic events in surface diffusion 3.1 3.2 3.3 3.4

4

page ix xi xiv

Adatom binding sites Atomic jumps in diffusion Long jumps in surface diffusion Transient diffusion

64 64 86 116 146

Diffusion on one-dimensional surfaces

183

4.1 4.2 4.3 4.4 4.5

184 186 193 194 200

Aluminum: Al(110), (311), (331) Nickel: Ni(110) Nickel: Ni(311) and Ni(331) Copper: Cu(110), Cu(311), Cu(331) Molybdenum: Mo(211)

vi

Contents

4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 5

Rhodium: Rh(110), (311), (331) Palladium: Pd(110), (311), (331) Silver: Ag(110), (311), (331) Tungsten: W(211) Tungsten: W(321) Iridium: Ir(110), (311), (331) Platinum: Pt(110), (311), (331) Gold: Au(110), (311), (331) Lead: Pb(110) Tables for 1D Diffusion

202 204 207 211 225 227 230 235 238 238

Diffusion on two-dimensional surfaces

261

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.30

261 265 269 269 274 277 283 295 301 304 304 309 312 315 322 328 329 341 344 347 348 353 355 361 365 368 372 372 373 402

Aluminum: Al(100) Aluminum: Al(111) Potassium Iron: Fe(100), (111), (110) Nickel: Ni(111) Nickel: Ni(100) Copper: Cu(100) Copper: Cu(111) Molybdenum: Mo(110), (111), (100) Ruthenium: Ru(0001) Rhodium: Rh(111), (100) Palladium: Pd(100) Palladium: Pd(111) Silver: Ag(111) Silver: Ag(100) Tantalum: Ta(110) Tungsten: W(110) Tungsten: W(100) Tungsten: W(111) Rhenium: Re(0001) Iridium: Ir(111) Iridium: Ir(100) Platinum: Pt(111) Platinum: Pt(100) Gold: Au(100) Gold: Au(111) Lead: Pb(111) Bismuth: Bi(111) Tables for 2D diffusion Predictions and comparisons

Contents

6

7

8

9

vii

Diffusion in special environments

423

6.1 6.2 6.3 6.4 6.5 6.6 6.7

423 430 479 485 487 498 506

Near impurities To descending step edges Atom lifetime versus the step-edge barrier Comparisons Atom descent over many-layered steps and between facets To ascending step edges Diffusion near dislocations

Mechanism of cluster diffusion

517

7.1 7.2 7.3 7.4 7.5

520 536 546 547 550

Via single atom jumps Concerted displacements Mechanism of dimer diffusion versus bond length Kinetic mechanisms of larger clusters Derivation of the mechanism of large cluster movements

Diffusivities of small clusters

556

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15

556 561 567 569 575 593 596 598 601 602 618 619 634 647 651

Early investigations Clusters on aluminum surfaces Clusters on iron surfaces Clusters on nickel surfaces Clusters on copper surfaces Clusters on rhodium surfaces Clusters on palladium surfaces Clusters on silver surfaces Clusters on tantalum surfaces Clusters on tungsten surfaces Clusters on rhenium surfaces Clusters on iridium surfaces Clusters on platinum surfaces Clusters on gold surfaces Comparisons

Diffusion of large clusters

664

9.1 9.2 9.3 9.4

664 676 692 693

Large clusters on fcc(100) surfaces Large clusters on fcc(111) surfaces Large clusters on fcc(110) surfaces Comments and comparisons

viii

Contents

10

Atomic pair interactions

696

10.1 Early measurements 10.2 More recent studies 10.3 Summary

698 704 730

Appendix: Preparation of samples for field ion microscopy Index

735 743

The color plates are situated between pages 298 and 299.

Preface

Surface diffusion on metals has been a subject of scientific interest for roughly ninety years. During the first forty years of this period it was very hard to do meaningful work because of technical problems – the difficulty of establishing good enough vacuum conditions to maintain a surface clean for measurements. In a few laboratories, mostly industrial, ultrahigh vacuum techniques were already practiced at that time, but this was not the normal course of events. All of this changed after World War II, first with the general adoption of good vacuum practices, and then with the development of more capable techniques for examining kinetic processes that are important on a surface. The first of these techniques was field ion microscopy, invented by Erwin Müller [1,2], the first method to provide a direct view of single atoms on a surface. The next important development was the scanning tunneling microscope, devised by Binnig and Rohrer [3], which established the capability of probing a large scale surface with high resolution. The last major contribution was the progress in theoretical techniques and computer technology, which toward the end of the twentieth century led to the rapid growth of theoretical calculations. The last forty years have therefore been a time of great progress in our understanding of surface diffusion, especially of metal atoms on metals. These advances have been spread over the scientific literature, and there has been no overview of the entire field, which is what we are trying to provide here. Our primary emphasis will be on experimental work to define the processes participating in surface diffusion. However, theoretical work can now be done so expeditiously that it has provided valuable guidance, and is now being intensively pursued. As such these contributions will also be carefully noted.1 Surface diffusion has, of course, a long history, dating back to the initiating work of Hamburger [5] in 1918. These early studies have, however, already been reviewed [6], so here we will be concerned with work on surface diffusion under ultra high vacuum (UHV) conditions and on an atomic scale, which began in the 1960s, and has led to the current state of understanding. The beginnings of modern studies of surface diffusion were greatly influenced by the insights and inspiration of David Turnbull, as well as by the traditions and expertise at General Electric. We have also benefited from the encouragement and suggestions of Ryszard Błaszczyszyn, and were able to draw on the expertise at the Institute of Experimental Physics of the University of Wrocław. Here, at the University of Illinois, 1

For a review of theoretical efforts, see T. Ala-Nissila et al. [4].

x

Preface

we have had helpful interactions with Dan Alpert, the man that guided the start of modern ultrahigh vacuum techniques which underlie diffusion studies on surfaces. Above all, GE wants to express his appreciation to his wife for her support and for the time devoted to this effort. The point of view of this presentation is primarily atomistic, and this was stimulated by the work of J. H. de Boer in his book The Dynamical Character of Adsorption, Clarendon Press, Oxford 1953, which had quite an impact on us. It is important to recognize that the term surface diffusion spans topics much broader than what we intend to cover here. Our concern will be concentrated on the behavior of single entities and clusters on a surface. This avoids encountering the interactions between atoms which affect surface diffusion at finite concentrations, and are specific to the particular chemistry of each system. However, with an understanding of surface diffusion gained from experiment and theory, work on interactions between adatoms will be described as well. Our efforts have greatly benefited from interactions with the various members of the Surface Studies group here over the years, and we express to them our great appreciation. We also want to emphasize again the crucial importance of experimental work, and of the technical support necessary for this. It is therefore a pleasure to give our thanks to the people who primarily provided this support for us: Bob Bales, Jack Gladin, William Lawrence, and Bob MacFarlane. Also important in coming to grips with the subject of surface diffusion was the assistance of Mary Kay Newman, the librarian in the Physics Department, whose help, as well as that of Nicholas Watanabe, has been really appreciated. Finally we want to acknowledge a special debt to Jennifer Lewis, who made it possible for us to continue our work.

References [1] E. W. Müller, K. Bahadur, Field ionization of gases at a metal surface and the resolution of the field ion microscope, Phys. Rev. 102 (1956) 624–631. [2] E. W. Müller, T. T. Tsong, Field Ion Microscopy Principles and Applications (American Elsevier, New York, 1969). [3] G. Binnig, H. Rohrer, Scanning tunneling microscopy, Helv. Phys. Acta 55 (1982) 726–735. [4] T. Ala-Nissila, R. Ferrando, S. C. Ying, Collective and single particle diffusion on surfaces, Adv. Phys. 51 (2002) 949–1078. [5] L. Hamburger, Ultra-microscopic examinations of very thin metal and salt deposits obtained by evaporation in a high vacuum, Kolloid Z. 23 (1918) 177–199. [6] G. Antczak, G. Ehrlich, The beginnings of surface diffusion studies, Surf. Sci. 589 (2005) 52–66.

Abbreviations

A AES AFW Ass ATVF A-Ex B CEM CEM59 CM Coh. COM COP CS CY-EAM CY-EAM1 CY-EAM2 DFT Diam DL D-Ex EAG EAM EAM5 EMT Ener min Eq. Ex FDB FEM FIM Fluct F-S

Type A step edge on fcc(111) Auger electron spectroscopy Adams, Foiles, Wolfer Assigned Ackland, Tichy, Vitek, Finnis Adatom catalyzed exchange Type B step edge on fcc(111) Corrected effective medium method CEM with 59 active atoms Concerted motion Cohesion approximation Center of mass Center of positions Constrained statics EAM of Cai and Ye Extension of CY-EAM Extension of CY-EAM Density functional theory Diameter Discommensuration line Double exchange Ercolessi–Adams glue potential Embedded atom method Embedded atom method 5 Effective medium theory Energy minimum Equation Exchange Foiles, Daw, Baskes Field electron emission microscopy Field ion microscope or microscopy Fluctuation Finnis–Sinclair

xii

List of abbreviations

GGA GP He-Scat 3 He- SE HR HRLEED K K K-K-R LAM LDA LDOS LEED LEEM LEIS L-Ex LF LMD L-J Mag MAEAM MC MD MD/MC-CEM MBE ML Morse MS MW M-Jump NEB nn Nucl OJ PACS PEEM Photo Pot RD Rean Refit Resis RGL

Generalized gradient approximation Glue potential Helium scattering 3 He spin echo High resolution High resolution low energy electron diffraction Kelvin Kink Korringa–Kohn–Rostoker method Lonely atom method Local density approximation Local density of states Low energy electron diffraction Low energy electron microsopy Low energy ion scattering Long exchange Leapfrog Langevin molecular dynamics Lennard Jones Magnetic Modified analytical embedded atom method Monte Carlo Molecular dynamics Molecular dynamics/Monte Carlo using CEM Molecular beam epitaxy Monolayers Morse potential Molecular statics Metastable walk Meta jump Nudged elastic band Nearest neighbor Nucleation theory Oh and Johnson Perturbed γ
γ angular correlation studies Photoemission electron microscope Photoemission Potential Ring diameter Reanalyzed Refitted and reanalyzed Resistivity Rosato, Guillope, Legrand

xiii

List of abbreviations

RHEED RS SC Scat SEAM SEM SI sim SPA-LEED Static STM T TB TDT TI TST T-Ex Q-Ex VASP VC VTST WF XPD XPS Z k ⊥

Reflection high energy electron diffraction Rutherford scattering Sutton–Chen Scattering Surface embedded atom method Scanning electron microscope Surface ionization Simulation Spot profile analysis of low energy electron diffraction Static barrier Scanning tunneling microscope or microscopy Temperature Tight-binding Tersoff, Denier van der Gon, and Tromp Thermodynamic integration Transition state theory Triple exchange Quadruple exchange Vienna ab initio simulation package Voter Chen Variational transition state theory Work function X-ray photoelectron diffraction X-ray photoelectron spectroscopy Band occupation In-channel Cross-channel

Symbols

α αfh =αhf αM αN =αL αRe a‘ aS A AR β βff =βhh βR βRe bS χc c c0 δ δF δD δxo δx =δy dd d12 dt dT dR D Do

Jump rate to nearest-neighbor position at the right for 1D motion, or jump rate to nearest-neighbor position for 2D motion Rate of single jumps from fcc to hcp/hcp to fcc site on fcc(111) Morse parameter Exponent describing dependence of diffusivity D on number of atoms N/on radius of island Rr, or island of length LL Rate of short range mechanism of movement for Re-Ir complex Lattice spacing Atom jump rate along step of type A Island area Parameter of repulsive energy Jump rate to next nearest-neighbor position at the right for 1D motion, or jump rate to next nearest-neighbor position for 2D motion Rate of double jumps between fcc/hcp sites on fcc(111) Jump rate for rebound jumps Long range mechanism of movement for Re-Ir complex Atom jump rate along step of type B Energy of condensation on fcc(111) plane Concentration, or rate of dimer jump via horizontal intermediate on bcc(110) Concentration at t = 0 Jump rate to nearest-neighbor position at the left in 1D movement Fermi-level phase shift Distance between interior and step edge barrier Kronecker delta Rate of horizontal/vertical jump on bcc(110) Distance Separation of atom 1 and 2 Rate of adatom motion on terrace Trio perimeter Plane diameter Diffusivity Prefactor of the diffusivity

List of symbols

D0B DM D205 Dγ D* ε εLJ ε1 =ε2 =ε3 εF εR εAA εxx/εyy e EA/EB st Esh 2 =E2 EiB ν ED D =ED i ER E‘0 E1 E2/ E3 E0 Eαf/Eαh Eα/Eβ/EβR Eδx/Eδy/Es Ea Eb Ecb Ecc Ecoh ECJ Ee/Eh Eeff B EA e =Ee B EA h =Eh Ei Ej =E‘ Eij Eij;‘

xv

Prefactor of diffusivity over descending step Morse parameter Diffusivity of cluster consisting of 205 atoms Diffusivity calculated with all types of jumps Prefactor in diffusivity dependence on cluster size Jump rate to next-nearest-neighbor position at the left in 1D movement Energy parameter of L-J potential First/second/third nearest-neighbor pairwise interaction Fermi energy Repulsive pair energy Interaction energy between two similar atoms at nearest-neighbor sites Strain Charge of the electron Activation energy for movement along step A/step B Barrier for dimer shearing / stretching Band energy Activation energy for movement obtained from diffusivity/velocity Repulsive energy between two atoms Energy of two adatoms at nearest-neighbor separation Energy of dimer in configuration 1, or binding energy for adatoms at nn separation Binding energy for adatoms in second/third nn separation Energy of dimer in configuration 0 Barrier height for jump out of fcc/hcp site Activation energy for single/double/rebound jumps Activation energy for vertical/horizontal/sum of jumps Additional step-edge barrier, or activation energy for jump a in dimer movement Activation energy for jump b in dimer movement Energy of core break up Energy of new row nucleation Cohesive energy Activation energy for concerted jump Barrier for exchange/hop Effective energy barrier Activation energy for exchange along step A/step B Activation energy for jump along step A/step B Cluster binding energy, or internal energy due to atom i Activation energy for j-/‘-type long jump Potential energy between atoms i and j Energy of two atoms at sites i and j in state ‘

xvi

List of symbols

Ek Ekd =Eku Eb‘ EB EB2 EHe ELF Ep Er ET Etot E(a,b) E(d) E(s) ΔE ΔEa ΔEb ΔEcs ΔED ΔEε ΔEe ΔEh ΔEint ΔEks ΔEkt ΔEvib ΔE(ε) 5E T 4 5ΔE4AT 5ΔE4COM  fij(Rij) ΔΦ fi(t) fj(Rij) F Fa Fe Ff

Energy of adatom pair in configuration k Energy of down jump/up jump at kink Activation energy for conversion from single to ‘-type long jump Activation energy for overcoming descending step and incorporate Activation energy for overcoming descending step at B2 position and incorporate Incident energy of helium Activation energy for leapfrog event Energy of movement along step Rebound energy Activation energy for diagonal transition around cluster corner Total energy Energy of atom pair at separation R = (a,b) Pair interaction energy at separation d Energy as a function of the displacement s Energy change Effective energy for movement of cross-channel dimer from state 0 to state 1 Effective energy for movement of cross-channel dimer from state 1 to state 0 Binding energy of core atom relative to adatom at step Energy of activation for diffusion Energy width in time of flight spectrum Activation energy for cluster movement by atom exchange Activation energy for cluster movement by atom hopping Interaction energy Binding energy of kink atom Binding energy of kink atom relative to atom on terrace Vibrational contribution to energy of activation Energy changes during collision Mean kinetic energy Effective activation energy for atomic motion of dimer Effective activation energy for center of mass motion of cluster Electron work function Core–core repulsion between atoms i and j Difference in structural energy between barrier peak and normal position Auto-correlation function for electron emission fluctuation Contribution of electron density of atom i arising from atom j Free energy Force Electric field Rate of atom deposition

xvii

List of symbols

Fx F(a,b) F(R) F(t) Fi ðρi Þ ΔF ΔFD γ γ
γ γs Γ Γo Γi Γε g g(R) G(t,z) h ha hc he hk hke/hse hr hre Hos
h i I Ie Iexp Ifit IR I I0 In(τ) j jB jD jR 

Free energy for atoms x units apart Free energy of atom pair at separation R = (a,b) Free energy of interaction as a function of the separation R Fraction of atoms on the surface Energy for embedding atom into local density ρi Free energy change Change in free energy for diffusion Jump rate to third neighbor position Angular correlation Formation energy per step atom Jump rate Prefactor for the jump rate Rate of dissociation of island of size i Quasielastic energy width of scattered atoms Geometrical factor Pair distribution function Moment generating function of variable z Planck’s constant Rate of detachment of atom adsorbed at straight edge Rate of core breakup Rate of straight edge hopping Rate of kink escape Rate of detachment of atom from kink/from straight edge to terrace Rate of conversion of vertical to horizontal dimer, or rate of corner rounding Rate of detachment of atom from corner to the step edge Enthalpy of sublimation h 2π Critical size of cluster Ionization potential Density of the emission current Intensity of scattered He atoms Best model fit to scattered He atom intensity Kinematic RHEED intensity Ratio of scattered to incident intensity Modified Bessel function of order n and argument τ Flux across unit length of line Atom jump rate over barrier EB at step edge Diffusive flux Flux at position R Ratio of force constants

xviii

List of symbols

k kh ka/ kke kk kforce kF λdeB λF λx ‘ ‘0 L L0 Li LL LT μ μx m ma m1, m2 M MS MT ν ν0 νs νh ν0α =ν0β =ν0βR να =νr =νce =νall ν0δx =ν0δy =ν0s ν0B νa νb νd ν‘ =νj νb‘ n nx nc nout/ nin

Boltzmann’s constant Harmonic approximation of escape rate Rate of atom attachment from terrace to straight step/to kink Rate of atom attachment from edge to kink Force constant Fermi wave number deBroglie wave length Fermi wave length Jump rate to right, starting from position x Jump length, or quantum state Nearest-neighbor spacing Number of sites in one-dimensional plane Standard length Island separation Island length or diameter also side length of square deposit Tip to detector distance Chemical potential Jump rate to left, starting from position x Mass of electron Number of deposited atoms, or number of atoms adsorbed per cm2 Number of atoms per unit length Number of atoms adsorbed, or total number of observations Number of surface sites Magnification of field ion microscope Attempt frequency of atom Frequency prefactor for diffusion Frequency prefactor for diffusion across descending steps Harmonic approximation attempt frequency for diffusing adatom Prefactor for single/double/rebound jumps Frequency of single/reinsertion/correlated / all jumps Prefactor for horizontal/vertical/sum of jumps Frequency factor for descending lattice step Relative frequency factor of step edge to terrace diffusion, or frequency for rate a in dimer motion Frequency for rate b in dimer motion Frequency factor Frequency factor for ‘-=j-type jumps Frequency factor for conversion from single to longer jump Number of jumps Number of islands per site Number of charges on the evaporated ion Number of paths for going out/in over boundary Number of diffusion events

List of symbols

N Na Nav =N0av NI/ NII Nc N f / Nh N i / Nt Nαf /Nαh NT N(R) No(R) N 5Δn21 4 p p(R) pb‘ p n1 px pΔxN P P0/P1 ðzÞ P0 P0A P1D/P2D Pb PE Pij P(N) ðfÞ ðhÞ Pf =Ph θ Θ q qc qF Qi ρi ρðtf Þ r

xix

Number of atoms in cluster, size of island, or total number of transitions (jumps) Number of atoms simulated Mean island density/initial post-deposition mean island density Frequency of occurrence of island in form I/form II Number of atoms in hexagonal form Number of atoms at fcc/hcp sites Number of atoms incorporated/trapped Number of hops out from fcc/hcp site to the same kind of site Total number of jumps Number of observations of two atoms separated by R Total number of atom pairs at separation R Average number of atom jumps Mean-square value of jumps to the right Probability of jump to the right Probablility of finding adatom pair at separation R Probability of converting from single to long jump Probability of reaching x ¼ s‘ after n1 jumps Probability of atom being at the distance x Number of atoms at displacement Δx Probability that material present at t = 0 will be gone at time τ Probability of being at a site of type 0/type 1 Probability of center of mass being at site of type 0 having started at z Probability of finding trimer in configuration 0A, regardless of position Probability of cluster in 1D/2D configuration Probability of atom overcoming step boundary Probability of atom occupy edge site Probability of finding a pair of atoms at sites i and j Term in prefactor for cluster diffusivity accounting for dynamical misfit Probablity of atom ending at fcc/hcp site when starting at the same kind of site Fractional occupation of sites Coverage Probability of jump to the left, or distance dependence of hopping integral Translational coordinate In-surface Fermi wave vector Desorption energy of ion Electron density of atom i Auto-correlation function Distance between dimer’s atoms, or rate of jumps at constant temperature

xx

List of symbols

r0 rc re ri req rT r1 R R jRj Ro Rb Rc RFi R‘ Rij Rr Rs RT Rx R1 5Δr24 σ σi σx s s0 s(T) S1/S0 s stot Sav =S0av ΔS ΔSD ΔSvib τ τ0 τc τf tf t

Rate of jumps during “zero-time” observations Rate of evaporation–condensation mechanism Rate of diffusion along cluster perimeter Rate of incorpotration to descending step Rigid distance between dimer’s atoms Tip radius Distance of descending step from center Atom deposition rate, or overall rate of jumping Adatom–adatom separation vector Adatom–adatom separation magnitude Morse parameter Rate of basic jumps, derived from low temperatures Cut-off distance for interactions Rate of field ionization Rate of long jumps of type ‘ Distance between atoms i and j Cluster radius or radius of circular deposit ED ΔHos Tip to screen distance Distance from the center of original distribution Distance of ascending step from center Mean-square displacement in 2D Fluctuation of displacement in 2D Interatomic separation at which potential energy vanishes Capture number, relating rate of incorporation to the diffusivity D L-J distance parameter Displacement from initial equilibrium Prefactor to s(T) Ratio of rate of step edge crossing to nearest-neighbor jumps on plane Entropy of dimer in configuration 1/in configuration 0 Relative distance Mean island size/initial mean island size Change in entropy of system Entropy of activation for diffusion Vibrational contribution to entropy of activation Mean lifetime for atom incorporation Prefactor for atom lifetime Lifetime for adatom starting at the center of plane Relaxation time for fluctuation Time interval for fluctuation Length of time interval

xxi

List of symbols

t0 tc te T Td TD TE TS Tm TR T*  e vA & v V V0 5v x4 ω0 ωd ω1/ω2/ω3 Ω ΩI/ΩII W ξ ξ 1 =ξ 2 5Δx24 5x24 X X(N) X y(Å) 5Δy24 5y24 zA Z ZT

Time interval for “zero-time” measurements Time interval for diffusion at constant temperature Slowly varying functions of 3.79 × 10–4Fe1/2/. Temperature Temperature for dissociation of cluster Temperature for diffusion Atom temperature Sample temperature Melting point Temperature for cluster rearrangement kT ε Correction term in field ionization Slowly varying function of 3.79 × 10–4Fe1/2/f Mean velocity in positive direction Effective hopping integral, or quarto interactions Velocity Voltage Effective barrier for non-interacting atoms Average x-component of velocity Angular Debye frequency Angular attempt frequency Frequencies Degeneracy Degenerency, number of equal configurations of form I/form II Free energy change between top and bottom of potential Mass of an incident compared with a lattice atom First/second trio interactions Fluctuation in displacement x Mean square displacement Mean diffusion length Overall displacement Pair separation measured along channel of W(211) plane Distance perpendicular to step but parallel to surface Fluctuation in displacement y Mean square displacement Partition function of adsorbed material Canonical partition function Tip sample distance

1

Atomic diffusion on surfaces

One important reason surface diffusion is of interest is that it is so different from diffusion in bulk solids, and is involved in many important processes – among them crystal and film growth as well as evaporation, chemical surface reactions, catalysis, and condensation. However, before becoming intimately involved in the description of surface events, it will be useful to outline the formalism that describes diffusion on surfaces.

1.1

Diffusivities: an introduction To describe diffusion on a crystal surface it is convenient to adapt the procedures developed for bulk diffusion [1]. The flux J crossing a line of unit length is given by Fick’s first law J ¼
D ∂c=∂x;

(1:1)

where ∂c/∂x gives the gradient of the concentration c and D is the diffusivity; the diffusivity establishes the magnitude of the flux in relation to the gradient, and is generally given in units of cm2/sec. Establishing a known gradient of the concentration on a surface, and measuring the flux J are difficult, and it is therefore useful to transform Eq. (1.1). Consider two parallel lines on a surface, shown in Fig. 1.1, a distance of Δx apart, which is comparable with the jump length ‘ executed in diffusion. The flux into line 1 will be different from that into line 2, as material accumulates in the region between the two lines. If the flux is considered per unit length, then J1 ¼ J2
∂J=∂xΔx:

(1:2)

The difference in the flux to the two lines can obviously be written as J1
J2 ¼
∂J=∂xΔx;

(1:3)

that is in terms of the amount of material accumulated, so that J1
J2 ¼
∂J=∂xΔx ¼ ∂c=∂tΔx:

(1:4)

However, from Eq. (1.1) we know that ∂J=∂x ¼
∂ðD ∂c=∂xÞ=∂x;

(1:5)

2

Atomic diffusion on surfaces

Δx

Γ Γ 1 Fig. 1.1

2

x

Schematic illustrating atomic jumps at rate Γ per atom in surface diffusion. Jump length = ‘.

and therefore ∂c=∂t ¼ ∂ðD ∂c=∂xÞ=∂x:

(1:6)

This is Fick’s second law, more directly applicable to examining surface diffusion. We will generally consider here diffusion of single atoms over a crystal plane, so that the diffusivity D will not be a function of the concentration, and Eq. (1.6) can therefore be written as ∂c=∂t ¼ D ∂2 c=∂x2 :

(1:7)

Note that we have only considered one-dimensional diffusion, with the flux as well as the gradient along the x-axis, but this will suffice for our problems. One possible way of looking at diffusion is to deposit a line of ma atoms on a surface, and to examine how the deposited material spreads out with time. The solution of Eq. (1.7) for this case is   ma x2 cðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi exp
; (1:8) 4Dt 4πDt where x is the distance normal to the initial deposit, and t indicates the length of the diffusion time interval; the solution is shown in Fig. 1.2a. Equation (1.8) can be readily confirmed by differentiating c(x,t) with respect to both x and t. The boundary conditions here are that at x = 0, c ! 1 and for |x| > 0 as t ! 0; c ! 0. If instead of just a line, a part of the entire crystal surface can be covered with adsorbed material, as in Fig. 1.2b; boundary conditions now are that with the border of material at x = 0, c = co for x > 0, and c = 0 for x < 0, both with t = 0. The covered region can be considered as an array of adjacent lines at a separation z. We just integrate the answer in Eq. (1.8) to give " # ð1 co ðx
zÞ2 cðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi dz: (1:9) exp
4Dt 4πDt 0 x
z If we let u ¼ pffiffiffiffiffiffiffiffi, then 4Dt

3

1.1 Diffusivities: an introduction

0.6

0.6

1

(a)

0.5

0.5

0.4

0.4

0.3

0.3

0.8

= 1.0 0.2

0.2

= 0.5

0 –2

0.6

0.2

0.1

0.1

(b)

= 0.5

Concentration

Concentration

= 0.0

–1 0 1 Displacement x

2

0 –3

3

–2

–1 0 1 Displacement x

2

3

Concentration profiles established in diffusion for different mean-square displacements. (a) Spreading from an initial straight-line deposit. (b) Spreading out of a half-covered surface.

co cðx; tÞ ¼ pffiffiffi π

ð pffiffiffiffi x

4Dt


1

  exp
u2 du:

Inasmuch as the error function is given by ð   2 x erfðxÞ ¼ pffiffiffi exp
u2 du; π 0 and erf(−1) = −1, the solution can be more simply written as    co x cðx; tÞ ¼ 1 þ erf pffiffiffiffiffiffiffiffi : 2 4Dt

(1:10)

(1:11)

(1:12)

Spreading will yield a gradually diminishing boundary region, as shown in Fig. 1.2b. An alternative geometry for observing atomic diffusion is to deposit a circular spot containing ma atoms and to study spreading from it. The solution to the diffusion equation then is   ma Rr 2 exp
; (1:13) cðRr ; tÞ ¼ 4πDt 4Dt where Rr gives the distance from the center of the original deposit. As a final example, consider the spreading of a deposit from the front of a ribbon of width d to the initially clean back. In this finite system we have the initial condition that at t = 0, c = co for 0 < x < d, and c = 0 for d < x < w, where w > d. The solution to the diffusion equation under these circumstances, given by Barrer [2], is  2 2 

d 2 X1 1 nπd nπx nπ þ sin cos exp
2 Dt : (1:14) cðx; tÞ ¼ co n¼1 n w π w w w

4

Atomic diffusion on surfaces

By measuring the spreading as a function of time and position, values of c(x,t) can be established experimentally. The diffusivity is then obtained by fitting the appropriate solution to the measured concentration profile. It is clear that here we have covered only the simplest examples useful in studies of surface diffusion. More complicated ones are described by Crank [1]. It should also be noted that under many conditions these approaches are not at all easy to implement. With a value of the diffusivity in hand, the question arises immediately how to interpret the diffusivity D in terms of the atomic jump processes. We follow here the approach pioneered by Einstein [3]. Consider a surface of unit width, with a uniform concentration gradient in the x direction. Atoms jump in the x direction at the rate Γ per atom, and in the –x direction at the same rate. We now draw two lines perpendicular to the x-axis as was done in Fig. 1.1; the second is separated from the first by a distance equal to the jump length ‘ executed by an atom in diffusion. The rate at which atoms cross line 1 is m1Γ, and line 2 is m2Γ where m1 and m2 are the number of atoms per unit length. The net rate at which atoms are transferred to the right will be given by m1 G
m2 G ¼ Gðm1
m2 Þ;

(1:15)

that is, the flow to the right from line 1 is compensated to some extent by the flow to the left from line 2. The number of atoms ma can be related to the surface concentration by ma = c‘, so that the net flux J becomes J ¼ ðc1
c2 Þ‘G:

(1:16)

Now c1 = c2 – ‘∂c=∂x, so that J ¼
‘2 G∂c=∂x:

(1:17)

From Eq. (1.1) it follows that D ¼ G‘2 ;

(1:18)

and we see that the diffusivity is just given by the product of the jump rate Γ per atom in one direction times the square of the jump length. For a more realistic view of diffusion, ‘ should of course be taken as the square root of the average of the squares of the individual displacements. For the jump rate Γ per atom that has entered here we can write the expression available from transition state theory for the rate of overcoming a potential barrier of height W [4],   W G ¼ ν exp
; (1:19) kT where ν accounts for the vibrational frequencies of the system, known also as attempt frequency. The diffusivity D can therefore be written as   W D ¼ ν‘2 exp
: (1:20) kT

1.1 Diffusivities: an introduction

5

Here W is really a free energy change ΔF between the top of the potential and the atom in its equilibrium position, but confined to a plane perpendicular to the diffusion path. Since ΔF = ΔE – TΔS, the diffusivity becomes     ΔSD ΔED exp
: (1:21) D ¼ ν‘2 exp k kT The usual approximation is that ν, ‘, and ΔSD are not strongly dependent upon the temperature T, so that a plot of ln (D) versus 1/T will provide us with ΔED , the activation   ΔSD 2 as the y-intercept energy for diffusion, as the slope and the logarithm of ν‘ exp k of the diffusivity D, with ΔSD as the change in the entropy in diffusion. It is customary to write   ΔSD (1:22) ¼ ν0 ν exp k as we often have some idea about the expected jump length ‘, and D can be more briefly written as   ΔED (1:23) D ¼ Do exp
; with Do ¼ ν0 ‘2 : kT where Do is known as the prefactor for the diffusivity. A somewhat different formulation has been offered by Kürpick et al. [5], who considered the transfer of an otherwise unconstrained atom from a normal site to the top of the barrier peak, where the degree of freedom in the direction of diffusion is withdrawn from the free energy. She arrived at an expression for the diffusivity as   Δ D ¼ Do exp
; (1:24) kT where Δ is the difference in the structural energy of the system between the barrier peak and the normal minimum. The prefactor, itself a function of the temperature T, is given by     kT 2 ΔSvib ΔEvib exp
: (1:25) ‘ exp Do ¼ k kT h Here ΔSvib and ΔEvib give the difference between the peak maximum and the minimum in vibrational contributions to the entropy and the internal energy, which have been evaluated by Kürpick [6]. It is useful to establish another connection between diffusivity and jump length, as the standard approaches for evaluating diffusivities are often difficult to carry out. We therefore evaluate the distance covered by a long sequence of N transitions, where N = 2Γt. If transition i gives a vector displacement xi, then the overall displacement x(N) will be XN x: (1:26) xðNÞ ¼ x1 þ x2 þ x3 þ    þ xN ¼ i¼1 i

6

Atomic diffusion on surfaces

On squaring the above we get XN XN
j XN
1 x2 ðNÞ ¼ x2 þ 2 xi xiþj : i¼1 i i¼1 j¼1

(1:27)

Averaging x2 ðNÞ causes the second term on the right to disappear: in a random displacement there is no relation between one jump and the next, so positive and negative transitions are equally probable. The mean-square displacement therefore becomes  2 XN 2 x ðNÞ ¼ x ¼ N‘2 ; (1:28) i¼1 i where the mean-square jump length ‘2 is given by ‘2 ¼

1 XN 2 x: i¼1 i N

(1:29)

That is, the mean-square displacement is just equal to the total number of jumps times the square of the jump length. The same result can be obtained in a slightly different way given by Berg [7] for a system of M particles. Consider the x-displacement after n jumps, xðnÞ; this is related to the displacement that has occurred previously by xi ðnÞ ¼ xi ðn
1Þ  ‘;

(1:30)

where ‘ is the length of the displacement. For the mean value of xðnÞ we obtain hxðnÞi ¼

1 XM 1 XM ½ x ð n
1 Þ  ‘  ¼ x ðn
1Þ ¼ hxi ðn
1Þi: i i¼1 i¼1 i M M

(1:31)

The mean location does not change as the number of steps changes, so that a particle starting at x ¼ 0 will remain there. The mean value hxðnÞi is therefore zero. For the square of the displacement we find x2i ðnÞ ¼ x2i ðn
1Þ  2‘xi ðn
1Þ þ ‘2 ;

(1:32)

and the mean-square displacement is given by  2 1 XN 2 1 XN x ð nÞ ¼ x ð n Þ ¼ ½x2 ðn
1Þ  2‘xj ðn
1Þ þ ‘2 : j j¼1 j¼1 j M M

(1:33)

However, the second term under the brackets at right disappears, as positive and negative terms balance out, so that  2  2 x ðnÞ ¼ x ðn
1Þ þ ‘2 : (1:34) When n = 0, x(n) = 0, so that x2 ð1Þ ¼ ‘2 , x2 ð2Þ ¼ 2‘2 , and x2 ðnÞ ¼ n‘2 . Therefore,  2 x ðnÞ ¼ n‘2 ; (1:35) as has already been demonstrated. We have previously shown in Eq. (1.18) that D = Γ‘2 ;  but hxðnÞi ¼ 0 and it follows from Eq. (1.28) that G‘2 ¼ x2 =2t, so that  2  2 Δx ¼ x
hxi2 ¼ 2Dt; (1:36)

1.2 Distribution of atomic displacements

7

the Einstein relation. In other words, the diffusivity can be derived directly from measurements of the displacement fluctuation or dispersion . We are also going to be interested in the atomic jump processes participating in diffusion. To learn more about these involves just a slight extension – looking in detail at the distribution of atomic displacements. This will be done in various ways in what follows.

1.2

Distribution of atomic displacements With diffusivities known from measured displacement fluctuations, what can we learn about the atomic jumps contributing to diffusion over a crystal surface? To pursue this question, we will adopt a more detailed view, and explicitly consider diffusion as arising from the movement of atoms or particles; this will involve us in a little more elementary mathematics.

1.2.1

Binomial distributions Assume that a particle makes a total of N independent, uncorrelated steps, each of length ‘ along an infinite straight line [8]. All jumps take place at the same time interval. We seek the probability px that the particle, having started at the origin, will end at position x ¼ s‘, but for the sake of simplicity we will assume the nearest-neighbor jump length to be unity. If the probability of a jump to the right is p and that for a jump to the left is q, then the likelihood of one configuration of n1 jumps to the right and N
n1 to the left is pn1 qN
n1 . Note that p þ q ¼ 1. How many different independent configurations are there for reaching the endpoint s‘? The first jump can be assigned in N ways on an empty line, the second in N − 1, and so on. In total we therefore have N! different choices. However, the selections for the n1 steps to the right all lead to the same result, and there are n1! different arrangements for such steps. The same can be said about the N − n1 steps to the left, which can be picked in (N − n1)! different ways giving the same effect. The total number of different configurations is therefore N! : (1:37) n1 !ðN
n1 Þ! The probability of reaching the point s after n1 steps to the right becomes p n1 ¼

N! pn1 qN
n1 : n! !ðN
n1 Þ!

(1:38)

From the binomial theorem we know that ðp þ qÞN ¼

N X n1

N! pn1 qN
n1 : !ðN
n Þ! n ! 1 ¼0

(1:39)

Since p + q = 1, it is clear that the normalization condition N X n1 ¼0

p n1 ¼ 1

(1:40)

8

Atomic diffusion on surfaces

is satisfied. The average number of jumps to the right is given by N X

hn1 i ¼

n 1 p n1

n1 ¼0

N! qN
n1 : n! !ðN
n1 Þ!

(1:41)

This is easy to evaluate if we remember that n1 pn1 ¼ p

∂ n1 p ; ∂p

(1:42)

so that hn1 i ¼

N X

p

n1 ¼0

∂ n1 N! qN
n1 : p ∂p n! !ðN
n1 Þ!

(1:43)

On interchanging the order of summation and differentiation, we have hn1 i ¼ p

N ∂ X N! qN
n1 : pn1 ∂p n ¼0 n! !ðN
n1 Þ!

(1:44)

1

Taking advantage of the binomial theorem we find that hn1 i ¼ p

∂ ðp þ qÞN ¼ Npðp þ qÞN
1 ¼ Np; ∂p

(1:45)

and for the average number hn2 i of jumps to the left hn2 i ¼ N
hn1 i ¼ Nð1
pÞ ¼ Nq:

(1:46)  2 For the mean of the second power of the number of jumps to the right, n1 , we proceed in an analogous fashion. h i  2 ∂ n1 ¼ p pNðp þ qÞN
1 ¼ p Nðp þ qÞN
1 þ pNðn
1Þðp þ qÞN
2 ∂p

(1:47)

and  2 n1 ¼ Npðq þ npÞ: For the fluctuation of n1 we therefore find E  2 D Δn1 ¼ ðn1
hn1 iÞ2 ¼ Npðq þ npÞ
N2 p2 ¼ Npq:

(1:48)

(1:49)

We now have many of the interesting quantities for the number of steps n1 to the right. The value of the position x with ‘o unity is given by x ¼ n1
n2 ¼ n1
ðN
n1 Þ ¼ 2n1
N:

(1:50)

If the number of steps n1 to the right is known we also know the position x. The distribution px of the position x is therefore the same as for jumps to the right, given by Eq. (1.38). The mean value of x becomes

1.2 Distribution of atomic displacements

hxi ¼ hn1 i
hn2 i ¼ Nðp
qÞ:

9

(1:51)

For the displacement we have Δx ¼ x
hxi ¼ 2n1
N
ð2hn1 i
NÞ ¼ 2ðn
hn1 iÞ ¼ 2Δn1 so that for the displacement fluctuation  2 Δx ¼ 4Npq:

(1:52)

(1:53)

The distribution can also be written more clearly in terms of the displacement x by taking advantage of Eq. (1.50). This gives Nþx N! N
x   pð 2 Þ qð 2 Þ : px ¼  Nþx N
x ! ! 2 2

(1:54)

When jumps to the right occur with the same probability as to the left this reduces to  N N! 1   px ¼  : (1:55) Nþx N
x 2 ! ! 2 2 We now have an expression that gives us the number of jumps N in terms of the measured displacements, and from Eq. (1.54) we can also find out the jump rates to the left and right.

1.2.2

Approximation for large values of N When the total number of jumps N becomes large, evaluation of the probability pn1 given by Eq. (1.38) requires more work, but an approximation can be reached readily. For large values of N, the probability pn1 at the maximum becomes large and n1 also assumes quite a large value. The condition for the maximum is readily derived by operating on the logarithm of the probability, which is less sensitive to n1 , lnðpn1 Þ ¼ lnðNÞ!
lnðn1 Þ!
lnðN
n1 Þ! þ n1 lnðpÞ þ ðN
n1 Þ lnðqÞ:

(1:56)

The necessary condition for the maximum is d lnðpn1 Þ ¼ 0: dn1

(1:57)

For large values of N, we can resort to Stirlings approximation lnðNÞ!  N lnðNÞ
N, so that d lnðNÞ! ¼ lnðNÞ dN

(1:58)

d lnðpn1 Þ ¼
lnðn1 Þ þ lnðN
n1 Þ þ lnðpÞ
lnðqÞ ¼ 0: dn1

(1:59)

and from Eq. (1.54) we find

10

Atomic diffusion on surfaces

It follows that ln

  ½ðN
5n1 4Þp ¼ 0; 5n1 4q

(1:60)

where 5n14 is the value of n1 at the maximum. Now ðN
hn1 iÞp ¼ hn1 iq

(1:61)

Np ¼ hn1 iðp þ qÞ

(1:62)

hn1 i ¼ Np;

(1:63)

so that

and, since p þ q ¼ 1,

which agrees with the result already obtained for the binomial distribution in Eq. (1.45). We can expand the logarithm of the probability pn1 around the maximum as lnðpn1 Þ ¼ lnðphn1 i Þ þ

d lnðpÞ 1 d 2 lnðpÞ 2 Δn1 þ Δn1 þ   : dn1 2 dn21

(1:64)

The derivatives are all evaluated at the maximum so that the first derivative vanishes, and for the second we get from Eq. (1.59) d 2 lnðpÞ 1 1 N : ¼

¼
2 n1 N
n1 n1 ðN
n1 Þ dn1

(1:65)

At the peak, hn1 i ¼ Np and N
hn1 i ¼ Nð1
pÞ ¼ Nq, so that d 2 lnðpÞ N 1 ¼
: ¼
NpNq Npq dn21

(1:66)

Note that the second derivative is negative as it must be at a maximum. Inserting the above result into Eq. (1.64) we obtain lnðpn1 Þ ¼ lnðphn1 i Þ
and

1 Δn21 2 Npq

  Δn21 pn1 ¼ B exp
; 2Npq

(1:67)

(1:68)

where B = phn1 i is just a constant of proportionality. Now the normalization requires that 1 ð pn1 dn1 ¼ 1 (1:69)
1

so we obtain 1 ð

B
1

  Δn21 dn1 ¼ 1: exp
2Npq

(1:70)

11

1.2 Distribution of atomic displacements

Substituting αk ¼

1 , we find after integration that 2Npq 1 B ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2πNpq

Finally we arrive at what is known as the Gaussian distribution   1 Δn21 pn1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp
: 2Npq 2πNpq

(1:71)

(1:72)

 We already have the average value of n1 . To find the fluctuation Δn21 , we note that 

E D 1 Δn21 ¼ ðn
hn1 iÞ2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πNpq

1 ð


1

  Δn21 ðn1
hniÞ2 exp
dn1 : 2Npq

(1:73)

Letting z ¼ Δn1 , we can write   z2 dz: z2 exp
2Npq

(1:74)

  pffiffiffi π z2 z2 exp
ð2NpqÞ3=2 ; dz ¼ 2 2Npq

(1:75)

 2 1 z ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πNpq

1 ð


1

The standard integral gives 1 ð


1

so that finally  2 Δn1 ¼ Npq:

(1:76)

The results here for the mean of n1 and the fluctuation of n1 is exactly the same as for the simple binomial with which we started. We already know from Eq. (1.52) that Δx ¼ 2Δn1 so that Eq. (1.72) for the distribution can be converted to   1 Δx2 ; (1:77) px ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp
8Npq 2πNpq and from Eq. (1.76) we obtain  2 Δx ¼ 4Npq:

(1:78)

We note here that just as in the binomial distribution, when the number of jumps N is even only even sites are populated, and when N is odd only odd sites are filled, inasmuch as Δx ¼ 2Δn1 .

1.2.3

Motion in continuous time In the presentation so far we have assumed that a fixed total number of jumps N is made and that these jumps take place regularly in time. Assume now instead that jumps may

12

Atomic diffusion on surfaces

ε

δ

α

β x

–4

Fig. 1.3

–2

0

2

4

Schematic of atomic processes for one-dimensional random walk. Nearest-neighbor jump length = ‘0 .

occur at any time, that the rate of jumping is constant, and that jumps occur randomly [9]. In order to anticipate future concerns, we are now also going to expand our horizons to allow not just jumps to nearest-neighbor sites to the right at the rate α and to the left at the rate δ, but also longer jumps spanning two nearest-neighbor distances [10–12]. Double jumps to the right occur at the rate β, and to the left at the rate ε, as indicated in Fig. 1.3. To simplify the presentation, we assume that the nearest-neighbor jump length ‘0 is unity. It is a simple matter to write out the equation for the time rate of change of the distribution function px as dpx ¼ βpx
2 þ αpx
1
ðα þ β þ δ þ εÞpx þ δpxþ1 þ εpxþ2 : dt

(1:79)

In order to proceed further, it is useful to introduce the moment generating function Gðt; zÞ, which is defined by 1 X

Gðt; zÞ ¼

z x px :

(1:80)

x¼
1

For the time derivative of this function we find :

Gðt; zÞ ¼

1

x  z

x¼
1

dpx ¼ dt

1

1

1

x¼
1

x¼
1

x¼
1

β  zx px
2 þ α  zx px
1
ðα þ β þ δ þ εÞ  zx px 1

1

x¼
1

x¼
1

(1:81)

þ δ  zx pxþ1 þ ε  zx pxþ2 : This can be written in a more convenient form as :

Gðt; zÞ ¼ βz2 þ

1 X x¼
1 1 X

zx
2 px
2 þ αz

1 X

zx
1 px
1
ðα þ β þ δ þ εÞ

x¼
1 1 X

δ ε zxþ1 pxþ1 þ 2 zxþ2 pxþ2 : z x¼
1 z x¼
1

A briefer way of presenting this equation is as   δ ε 2 _ Gðt; zÞ ¼ Gðt; zÞ
ðα þ β þ δ þ εÞ þ αz þ þ βz þ 2 : z z

1 X

z x px

x¼
1

(1:82)

(1:83)

13

1.2 Distribution of atomic displacements

The generating function is now obtained by integrating over time. Assuming that at time t = 0 the atom starts at x = 0, so that G(0, z) = 1, we find   h δ ε i Gðt; zÞ ¼ exp½
ðα þ β þ δ þ εÞt exp αz þ t exp βz2 þ 2 t : (1:84) z z The moments of the x coordinate are immediately accessible through  n 1 X ∂ n n x px ¼ z Gðt; zÞz¼1 : hx i ¼ ∂z x¼
1

(1:85)

The average displacement hxi is given by hxi ¼ ½α þ 2β
ðδ þ 2εÞt and for the mean-square value we find  2 x ¼ ðα þ 4β þ δ þ 4εÞt þ ðα þ 2β
δ
2εÞ2 t2 :

(1:86)

(1:87)

Third and fourth moment are given by:  3 x ¼ fα þ 8β
δ
8ε þ 3½ðα þ 2βÞðα þ 4βÞ
ðδ þ 2εÞðδ þ 4εÞ
2ðβδ
αεÞt þ ðα þ 2β
δ
2εÞ3 t2 gt

(1:88)

5x44 ¼ fα þ 16β þ δ þ 16ε þ 12ðβ þ δ þ 3εÞ2 t þ 12ðα þ 2β
δ
2εÞð3β þ δ þ εÞt þ ðα þ 2β
δ
2εÞ2  ½7 þ 6ðα þ 2β
δ
2εÞt þ 12ðβ þ δ þ 3εÞt þ ðα þ 2β
δ
2εÞ2 t2 tgt:

(1:89)

The fluctuations in x, = < x2> − < x >2 are therefore 5Δx24 ¼ ðα þ 4β þ δ þ 4εÞt:

(1:90)

When jumps to the right take place at the same rate as to the left, as they usually do in diffusion, then hxi ¼ 0

(1:91)

 2 Δx ¼ 2ðα þ 4βÞt:

(1:92)

5x34 ¼ 0

(1:93)

5Δx4 4 ¼ 2ðα þ 16βÞt þ 12ðα þ 4βÞ2 t2 :

(1:94)

Finite values of the odd moments indicate that motion is asymmetric. In Fig. 1.4 is illustrated the average displacement (first moment) together with the third moment of the displacements against the ratio of jump rates to the left compared to the right; here α and β are both equal to 1/sec, δ as well as ε are equal to the ratio of left to right. As the rate of

14

Atomic diffusion on surfaces

1.6

50

40

1.2 Average 1.0

30

0.8 20

0.6 0.4

Third Moment

Average Displacement

1.4

Third Moment 10

0.2 0 0.4

Fig. 1.4

0.5

0.6 0.7 0.8 0.9 1.1 Jump Rates Left/Right

0 1.1

Average as well as third moment of the displacement, as a function of the ratio of jump rates to the left compared with rates to the right. Diffusion time is 1 sec; rates α and β are set at 1/sec, δ as well as ε are equal to the ratio Left/Right. Error bars are estimates for 1000 observations of the third moment.

jumps to the left decreases, the average displacement increases, as the rates to the left are subtracted from rates to the right in Eq. (1.86). The third moment behaves in much the same way, but increases much more rapidly. Estimates of the standard error of the third moment characteristic for 1000 observations are small, so that a finite value for the third moment gives a clear indication of asymmetric movement. In the simplest case, where only single jumps occur, as in the previous sections,  2 Δx ¼ 2αt: (1:95) Deriving the average values of the displacement has been straightforward. Getting the distribution function for the displacements is more involved. As a start in this direction, we rewrite the generating function G(t,z) in Eq. (1.84) as " rffiffiffi!# rffiffiffi pffiffiffiffiffi α 1 δ Gðt; zÞ ¼ exp½
ðα þ β þ δ þ εÞt exp t αδ z  þ δ z α " !# rffiffiffi pffiffiffiffiffi 2 β 1 rffiffiεffi (1:96) þ : exp t βε z ε z2 β At this stage it is useful to recall the Schlömilch [13] relation    1 X τ 1 uþ ¼ exp un In ðτ Þ; 2 u n¼
1

(1:97)

where In ðτ Þ is the modified Bessel function of order n and argument τ. With this relation we are now able to carry out two transformations:

15

1.2 Distribution of atomic displacements

"

rffiffiffi!# rffiffiffi 1

α2i pffiffiffiffiffi X pffiffiffiffiffi α 1 δ exp t αδ z zi Ii 2t αδ þ ¼ δ z α δ i¼
1 and

"

pffiffiffiffiffi exp t βε z2

rffiffiffi  2j rffiffiffi!# 1 X pffiffiffiffiffi β 1 ε 2j β z Ij 2t βε : þ 2 ¼ ε z β ε j¼
1

(1:98)

(1:99)

The generating function can therefore be written as Gðt; zÞ ¼ exp½
ðα þ β þ δ þ εÞt 1 1

pffiffiffiffiffi pffiffiffiffiffi α2i β2j X X iþ2j  z Ii 2t αδ Ij 2t βε : δ ε i¼
1 j¼
1

(1:100)

With the substitution of x ¼ i þ 2j

(1:101)

the generating function transforms to  2j

pffiffiffiffiffi pffiffiffiffiffi αx
2j β 2 z Ix
2j 2t αδ Ij 2t βε : Gðt; zÞ ¼ exp½
ðα þ β þ δ þ εÞt δ ε x¼
1 j
1 1 X 1 X

x

(1:102) From the definition of the moment generating function in Eq. (1.80) we therefore arrive at the probability of finding a displacement x as j 1 x
2j  2 pffiffiffiffiffi

pffiffiffiffiffi X α 2 β px ¼ exp½
ðα þ β þ δ þ εÞt Ij 2t βε Ix
2j 2t αδ : (1:103) δ ε j¼
1 When the movement is asymmetric this is clear from the distributions, shown in Fig. 1.5. As the rate of jumps to the left decreases with respect to the jumps to the right, the distribution starts to be asymmetric and gradually is shifted to the right. A diminution of 10 percent in the ratio of jumps to the left with respect to the jumps to the right can be detected. When jumps to the right and left occur at the same rate, so that α ¼ δ and β ¼ ε, the distribution simplifies to 1 X Ij ð2βtÞIx
2j ð2αtÞ: (1:104) px ¼ exp½
2ðα þ βÞt j¼
1

If only nearest-neighbor transitions are allowed, the modified Bessel function Ij ð2βtÞ becomes zero for all orders except zero; in that order it is unity, and the probability can be written as px ¼ expð
2αtÞIx ð2αtÞ:

1.2.4

(1:105)

Comparisons The distributions presented here differ significantly. If the mean-square displacement for the binomial or the Gaussian distribution is odd then movement will occur only over odd

16

Atomic diffusion on surfaces

0.16 0.14

Ratio L/R = 1.0

L/R = 0.9

Probability p

x

0.12 0.1 0.08 0.06 0.04 0.02 0

–10 –8 –6 –4 –2 0

2

4

6

8 10

Displacement x

–10 –8 –6 –4 –2 0

2

4

6

8 10

6

8 10

6

8 10

Displacement x

0.16 0.14

L/R = 0.8

L/R = 0.7

Probability p

x

0.12 0.1 0.08 0.06 0.04 0.02 0 –10 –8 –6 –4 –2 0 2 4 Displacement x

6

8 10

–10 –8 –6 –4 –2 0 2 4 Displacement x

0.16 0.14

L/R = 0.6

L/R = 0.5

Probability p

x

0.12 0.1 0.08 0.06 0.04 0.02 0

Fig. 1.5

–10 –8 –6 –4 –2 0 2 4 Displacement x

6

8 10

–10 –8 –6 –4 –2 0 2 4 Displacement x

Distribution of displacements for an asymmetric random walk in one dimension. Rates α and β to the right are maintained constant at 1/sec; diffusion interval is always 1 sec.

sites; when the mean-square displacement is even then only the even sites are ever occupied. A random walk in continuous time, however, will populate all kinds of sites. This is most readily seen in plots of the probabilities with single jumps occurring at the same rate right and left, which are shown in Fig. 1.6. It is clear that a measurement of the distribution of probabilities should therefore reveal which expression describes the migration. If only every second site is ever found occupied, then the binomial or the Gaussian distribution is appropriate. However, if the atoms are observed at every site regardless whether they are even or odd then the continuous time description is pertinent.

17

1.3 Jump rates

0.4

0.4 (a)

Binomial

(b)

Gaussian 0.3 Probability px

Probability px

0.3

0.2

Fig. 1.6

0.2

0.1

0.1

0

Continuous

–6

–4

–2 0 2 Displacement x

4

6

0

–6

–4

–2 0 2 Displacement x

4

6

Comparison of binomial and Gaussian distributions in (a) with continuous time distribution in (b), all with the number of jumps equal to 4, and p = q = 1/2.

By quantitatively fitting the appropriate distribution to the experimental data, the rates of jump processes participating in the diffusion can be found. The asymmetry of movement, if it occurs, can also be detected from distribution studies. Of course much more data are required to attain a reasonable distribution than just for diffusivities, but this provides the information to determine the rates of atom jumps.

1.3

Jump rates In previous discussions we have introduced jump rates without any further examination of what they depend upon and how to formulate them. These are matters that we will now examine in more detail, relying on the transition state theory of reaction rates [14,15]. In diffusing over a crystal surface, an atom will experience a potential minimum at the equilibrium sites, which are separated by barriers that tend to localize the atoms. A schematic of such a one-dimensional potential is shown in Fig. 1.7. Of course the lattice is not static; atoms vibrate around their equilibrium positions and the barrier changes during such vibrations. The rate of jumping is given as the product of two terms: the probability that an atom will be at a value of the coordinate qc at the top of the barrier, divided by its lifetime. The probability of being at the top of the potential barrier we will write as ϖðqc Þδqc , and the lifetime in this activated state is δqc =v A . Here v A is the mean velocity in the positive direction and qc is the distance along the path of least potential energy in moving from one site to another, usually referred to as the reaction coordinate. To describe the system in the simplest terms we assume that atoms at the top of the barrier are in a separate state, the transition state, in equilibrium with the normal atoms at

18

Atomic diffusion on surfaces

ΔED

Fig. 1.7

Schematic of potential energy of an adatom undergoing diffusion along a line on a perfect surface.

the equilibrium sites. This of course means that the chemical potentials in the two states are the same, μA ¼ μN ;

(1:106)

where the subscript A designates the active or transition state and N the normal or equilibrium state. For a localized material of non-interacting particles we have μ z0 ¼
ln
lnð1
θÞ; θ kT

(1:107)

where z0 is the partition function referred to a standard energy reference. The fractional occupation of surface sites, θ, is given by the ratio of the number of atoms to surface sites, that is by M=MS. We are interested in the diffusion of individual atoms, so that θ will be vanishingly small and can be neglected compared to unity. The equality of chemical potentials therefore leads to z0A z0N ¼ : (1:108) θA θN We can write this equation as   θA zA ΔED ¼ exp
; (1:109) θ N zN kT where the partition functions are now evaluated with respect to the bottom of their potentials and ΔED indicates the height of the energy barrier; that is the difference in energy between the bottoms of the two potential curves. For one-dimensional motion, the number of sites in the activated state is equal to the number in the equilibrium state, so that   MA zA ΔED : (1:110) ¼ exp
MN zN kT We can also write out the probability of being at the top of the barrier, $ðqc Þδqc ¼

MA MA  ; M A þ MN M N

(1:111)

where MN 44MA . To calculate the rate of moving from the top of the barrier, we assume that all atoms in a small length δqc of the coordinate qc at the saddle are in the activated state. The length δqc must be so small that the curvature of the potential function is unimportant and motion

19

1.3 Jump rates

can be considered as a classical translation. Note, however, that δqc must be large in comparison with the de Broglie wavelength λdeB , where λdeB

 1 2 h h2  ¼ : 2πmkT mv

(1:112)

This condition insures classical behavior for this degree of freedom. We now assume that the partition function for the activated state can be given by the product of the classical partition function for translation, ztrans , times a contribution from the remaining degrees of freedom, z# . That is, zA ¼ z# ztrans ¼ z#



2πmkT h2

1 2

δqc :

(1:113)

From Eq. (1.110) the number of atoms MA can now be written as  1   z# 2πmkT 2 ΔED : δq exp
MA ¼ M N c zN kT h2

(1:114)

To find the rate of diffusion, MA , the number of atoms in the activated state, must be multiplied by the rate at which an atom leaves the activated state to the right. We take this rate v A as the mean absolute value of the velocity 1 v A ¼ hjvji 2

(1:115)

  mv 2 dv jvj exp
2kT
1 hjvji ¼ 1   : Ð mv 2 exp
dv 2kT
1

(1:116)

and 1 Ð

Substituting a ¼

m , we find 2kT 1 ð   1 jvj exp
av 2 dv ¼ a

(1:117)


1

and 1 ð

  exp
av 2 dv ¼


1

rffiffiffi π ; a

(1:118)

so that 1 hjvji ¼ a

rffiffiffi  1 a 2kT 2 ¼ : π πm

(1:119)

20

Atomic diffusion on surfaces

Only half of the atoms will be jumping in the right direction, so that the rate at which a single atom moves, the rate constant, is given by   1 ΔED hjvji kT z# ¼ exp
G ¼ MA : 2 δqc h zN kT

(1:120)

To illustrate this result, we evaluate the rate for a simple, if unphysical model: the motion of an atom over a rigid one-dimensional potential. In the partition function for the activated state we no longer consider contributions in the direction of diffusion; only the motion in the normal state remains in zN , so that   hν   1
exp
z# 1 hν kT   ¼ 2 sinh ¼ ¼ ; hν zN zvib 2kT exp
2kT

(1:121)

here ν is the frequency of vibration of an atom in the normal state. Expanding the hyperbolic sine, we find 

hν 2 sinh 2kT Provided





hν kT

¼

# "   1 hν 2 þ : 1þ 24 kT

hν 51, the jump rate Γ in this example becomes kT     kT hν ΔED ΔED G¼ ¼ ν exp
: exp
kT kT h kT

(1:122)

(1:123)

The rate of jumping is given by the product of ν, the frequency of vibration multiplied by a Boltzmann term involving the barrier height ΔED . As already mentioned, this is not a realistic model. However, Vineyard [4] has shown that the jump rate can be quite generally written as n Q

νj

j¼1

G ¼ n
1 Q

νi



 ΔED : exp
kT

(1:124)

i¼1

In the prefactor we have the ratio of the product of n normal frequencies of the system to the product of n − 1 frequencies when the atom is located at the saddle point. At this stage it should be emphasized that the dimensionality of the partition functions in Eq. (1.121) is not the same – in the normal state it is higher by one than in the activated state. For a classical system, this is easily changed. We rewrite the partition function for normal atoms as zN ¼ z1 zvib :

(1:125)

21

1.3 Jump rates

where z1 is the partition function with the one degree of vibration removed. If the vibrational contribution zvib is assumed to be classical, then it follows that zvib 

kT hν

(1:126)

and     kT z# hν ΔED z# ΔED G¼ ¼ ν exp
: exp
kT z1 kT h z1 kT

(1:127)

We now have an expression for the jump rate Γ that is useful if we can evaluate the two partition functions. It is more convenient for our applications, however, to make a thermodynamic transformation. The standard free energy change ΔFD in going from the normal state (with one vibration removed) to the activated state is given by  #   ΔFD z ΔED exp
¼
ln : (1:128) kT z1 kT The jump rate Γ for diffusion can therefore be written as        ΔFD ΔSD ΔED ¼ ν exp exp
; G ¼ ν exp
kT k kT

(1:129)

where ΔSD gives the entropy and ΔED the energy of activation. Here it is customary to set the prefactor Γ0 in the jump rate to   ΔSD (1:130) G0 ¼ ν exp k and   ΔED G ¼ G0 exp
: kT

(1:131)

The displacement fluctuation is therefore  2  (1:132) Δx ¼ 2Gt ‘2 ;  2 where ‘ is the mean-square  value of the jump lengths. From the Einstein relation, Eq. (1.36), we know that Δx2 ¼ 2Dt. It therefore follows that the diffusivity D is given by  D ¼ G ‘2 ; (1:133) that is, the diffusivity D is just equal to the product of the jump rate and the mean-square jump length. We have so far concentrated on one-dimensional motion. Little new enters when motion is two-dimensional.  2  2 Consider diffusion by nearest-neighbor jumps on a (100) plane, where Δx ¼ Δy . Since  2  2  2 (1:134) Δr ¼ Δx þ Δy

22

Atomic diffusion on surfaces

χ

β

3a

α α

2

β

η Fig. 1.8

Schematic of atom arrangement on bcc(110) plane, showing jumps to first and second nearestneighbor sites along .

we now can write

 2 Δr ¼ 4Dt:

(1:135)

On a bcc (110) plane, however, this simple relation does not hold, since only in atomic units is 5Δx24 ¼ 5Δy24. We therefore again assume that movement occurs by nearest-neighbor jumps but along the close-packed directions , that is along the χ and  coordinates in Fig. 1.8. The relation between coordinates now is x¼χþ

y ¼ χ
:

Diffusion along χ and  axes is not correlated, so that  2  2  2  2 Δx ¼ Δy ¼ Δχ þ Δ and

 2 Δx ¼ 4αt;

(1:136)

(1:137)

(1:138)

with the unit of  length being a‘ =2, that is half the lattice constant; α is the rate of jumps along . Δy2 can be described by the same relation  2 (1:139) Δy ¼ 4αt; pffiffiffi but with the unit of length being equal to a‘ 2=2. On the fcc(111) plane the situation is more complicated, as atoms may sit in either fcc or hcp sites, indicated in Fig. 1.9. Atom jumps, shown in the same figure, occur at the rate αfh from fcc to hcp sites and at the rate αhf for the reverse process. Expressions have been worked out [10] relating the diffusivity to the jump rates,  αfh αhf 2 ‘ =2; D ¼ Δx2 =2t ¼ αfh þ αhf 0

(1:140)

where ‘0 is the nearest-neighbor spacing. Evaluation of the diffusivity D in terms of jump processes is therefore straightforward for both one- and two-dimensional processes.

23

References

hcp Sites

fcc Sites x+1,y+1

x–1,y+1

αhf

x–2,y

2

x–3,y+1

5

9

6

8

x–3,y+1

βff x+3,y–1

x–3,y–1

αfh

x+1,y–1

x,y+2

x+3,y+1

4

x,y–2

2

3

x,y+2

βhh

Fig. 1.9

x–2,y

3

x–1,y–1

x–3,y–1

1

1

4 9 8

x+3,y+1

5 6 7

x+3,y–1

x,y–2

Atom arrangement on fcc(111) plane, showing short as well as long jumps from fcc and hcp sites [10].

References [1] J. Crank, The Mathematics of Diffusion, 2nd edn. (Clarendon Press, Oxford, 1975), Chapter II. [2] R. M. Barrer, Diffusion in and through Solids (Cambridge University Press, London, 1941), p. 14. [3] A. Einstein, Elementare Theorie der Brownschen Bewegung, Z. Elektrochem. 14 (1908) 235–239. [4] G. H. Vineyard, Frequency factors and isotope effects in solid state rate processes, J. Phys. Chem. Solids 3 (1957) 121–127. [5] U. Kürpick, A. Kara, T. S. Rahman, Role of lattice vibrations in adatom diffusion, Phys. Rev. Lett. 78 (1997) 1086–1089. [6] U. Kürpick, Self-diffusion on (100), (110), and (111) surfaces of Ni and Cu: A detailed study of prefactors and activation energies, Phys. Rev. B 64 (2001) 075418 1–7. [7] H. W. Berg, Random Walks in Biology (Princeton University Press, Princeton, 1983), p. 6–11. [8] S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943) 1–89. [9] G. Ehrlich, Atomic displacements in one- and two-dimensional diffusion, J. Chem. Phys. 44 (1966) 1050–1055. [10] J. D. Wrigley, M. E. Twigg, G. Ehrlich, Lattice walks by long jumps, J. Chem. Phys. 93 (1990) 2885–2902. [11] F. Montalenti, R. Ferrando, Universal law for piecewise dimer diffusion, Phys. Rev. B 60 (1999) 11102–11109. [12] G. Antczak, G. Ehrlich, Asymmetric one-dimensional random walks, J. Chem. Phys. 129 (2008) 124702 1–4. [13] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chapter 9. [14] S. Glasstone, K. J. Laidler, H. Eyring, The Theory of Rate Processes (McGraw-Hill Book Co., New York, 1941). [15] K. J. Laidler, J. H. Meiser, Physical Chemistry, 3rd edn. (Houghton Mifflin, Boston, 1999).

2

Determination of adatom movements

Surface diffusion studies on single adsorbed entities, the focus of our presentation, had to await the development of techniques capable of revealing atoms. This was first accomplished by Müller [1,2] roughly fifty years ago in 1956, with his invention of the field ion microscope (FIM). The natural extension of FIM was the development of the Atom Probe [3] which allowed identification of chemical identities and control of composition for surfaces, but there also were earlier investigative methods, such as field emission microscopy [4], helium scattering [5], contact potential measurements [6] and so on, which provided useful information about surface diffusion. Today there are newer techniques that have been shown to have the capability of revealing atoms. The scanning tunneling microscope (STM), devised by Binnig and Rohrer [7,8] in 1983 is one of them. Less frequently used techniques, such as measurements of work function changes, perturbed γ
γ angular correlation, or atomic beam scattering will also be mentioned, if only very briefly. Insights into diffusion phenomena on the atomic scale gained with the scanning tunneling microscope are certain to grow in number and importance. Both field ion [9–11] and scanning tunneling microscopy [12–15] have been covered extensively in the literature, and will also be described here in reference to diffusion studies. It should be noted that for examination of diffusion phenomena on clean surfaces by any of these techniques, good vacuum conditions are crucial. The influence of just small amounts of impurities on the movement of atoms, even in the range of 10−10 Torr, has recently been recognized [16]. The microscopes must therefore all be built in such a way as to allow bakeout at temperatures of 300 °C or higher, and careful procedures for outgassing must be followed.

2.1

Field ion microscopy

2.1.1

Imaging equipment A picture of a field ion microscope suitable for examining atom diffusion is shown in Fig. 2.1 [17]. The vital part of the microscope is the tip, welded onto the top of the support loop and positioned opposite to the phosphor screen. An image of the surface is obtained by admitting an imaging gas (helium is the best candidate, but neon can be used as well), at a pressure of 80 K it can still move, and populates adjacent sites at the boundary with the empty zone, as shown in Fig. 3.36a. The central site is most frequently populated and from the time spent in the center compared to adjacent sites the difference in energy was derived as 0.015±0.002 eV. This suggests that the transition from center to the empty zone is gradual. This gradual transition is not limited to

86

Atomic events in surface diffusion

Fig. 3.37

Schematic of potential seen by Pt adatom diffusing radially on Pt(111) cluster [26].

small clusters; a diminishing population is also seen in Fig. 3.36b, for a cluster with a diameter of 16 nearest neighbors. For platinum atoms on Pt(111), an estimate of the potential energy changes over the surface is given in Fig. 3.37. Although the characteristics of atom motion in the empty zone are not well established, the potential energy in this region is definitely higher. The cause for the empty zone has not been clearly established, but may be due to surface stress, which is high for Pt(111). The cluster atoms may relax their positions, so that the strain at the center may differ from that at the edges. The phenomenon of empty zones is not limited to platinum; it has also been observed for Ir(111) [37,38]. As appears from Fig. 3.38, the empty zone for iridium atoms on Ir(111) is again ~ 3 spacings wide, and does not vary sensitively with cluster size. For a cluster with a width of 7 spacings shown in Fig. 3.38b, only the central few sites are occupied after diffusion. When the size is reduced to 5 spacings in Fig. 3.38c, the empty zone covers the entire cluster and there are no equilibrium sites of low energy left; all the cluster sites are occupied by atoms after diffusion. Recently, an empty zone was also suggested present on the Cu(111) surface [39]. The empty zone is not a general phenomenon, however. On tungsten, the entire (110) plane up to the edges is filled by atoms diffusing at 340 K, as is clear in the diagram in Fig. 3.39. It must be recognized, of course, that in diffusivity measurements, only observations in the central region of a plane provide information about movement on the flat surface. This survey clearly shows that the presence of steps can influence the adatom potential landscape in a number of ways, and this influence does not have to be short ranged.

3.2

Atomic jumps in diffusion At the time of the first observations of individual atom movement in the field ion microscope, in the years 1966 to 1970, diffusivity was seen as a simple phenomenon. Adatoms were believed to jump randomly between nearest-neighbor sites on the surface. This view was consistent with what was emerging from experimental measurements of the temperature dependence of the diffusivity D; these yielded diffusivity prefactors Do of roughly 10−3 cm2/sec. From Eqs. (1.22) and (1.23) it appears that the prefactor is given by

3.2 Atomic jumps in diffusion

Fig. 3.38

87

Location of Ir adatoms on Ir(111) clusters of different sizes [38]. Solid circles give position of adatoms, open ones indicate edge atoms. Shown is grid for hcp sites. (a) Distribution on cluster with diameter of 19 spacings after diffusion at 120 K. (b) Ir distributions over Ir61 after equilibration at 120 K. (c) Ir adatoms cover entire Ir37 cluster, with a diameter of only 5 spacings, after equilibration at 115 K.

Do ¼ ν‘2 expðΔSD =kÞ;

(3:5)

where ΔSD is the entropy of activation, ν the attempt frequency of the adatom, and ‘ the jump length. If we ignore the entropy of activation and pick a typical value of 1012 sec−1 for the vibrational frequency, then Do emerges on the order of 10−3 cm2/sec, as the nearest-neighbor jump length on metals is around 3 Å. These rough estimates suggested that this simple picture was correct in describing the movement of adatoms on the surface. Detailed information about surface diffusion dimensionality on tungsten, and later on rhodium surfaces, also fit well into this simple picture. On planes such as W(211) and W(321) shown in Fig. 3.1, as well as Rh(110), Rh(311), and Rh(331), in Fig. 3.2 , diffusion was one-dimensional and occurred along the channels that characterize these surfaces. This is evident, for example, from a detailed plot of atomic positions seen for Rh during diffusion on W(211), in Fig. 2.15. A look at the hard-sphere surface models

88

Atomic events in surface diffusion

Fig. 3.39

Location of tungsten adatom on W(110) plane after equilibration at 340 K [38]. Black dots indicate tungsten adatom, open circle shows edge atoms of W(110) plane. Entire surface is available for adatoms.

makes it clear the jumps over the sidewalls of the channels would require much more energy than along the direction of the channels, which should be the preferred path. This simple view, of being able to predict the directionality in surface diffusion merely by looking at an atomic model, did not, however, have a long lifetime.

3.2.1

Diffusion mechanisms: atom exchange on fcc(110) planes The simple picture of diffusion was destroyed by the work of Bassett and Webber [40], who in 1978 examined diffusion of platinum, iridium, and gold atoms on platinum surfaces. The most interesting results were obtained on the (311), (331), and (110) planes of platinum. On the former two, diffusion occurred as expected along the channels. On the (110) plane, however, diffusion was observed not only along the channels but also across them, making the diffusion two-dimensional. For diffusion along the channels, Bassett and Webber found an activation energy of 0.84±0.1 eV, and a prefactor of 8 × 10−3 cm2/sec, whereas motion across the channels required an activation energy of 0.78±0.1 eV and a prefactor of 1 × 10−3 cm2/sec. For iridium atoms, comparable results were found, but gold diffused along the channels of the platinum substrate. To explain the unexpected two-dimensional motion, Bassett and Webber considered possibilities such as impurities, but finally opted for two models. Fluctuations could occur in the location of the atoms making up the channel walls, opening up holes through which the adatom could move into an adjacent channel. A more likely alternative was that a fluctuation caused an atom from the channel wall to move aside, as in Fig. 3.40, allowing an adatom to move into the empty site; the atom from the lattice then took its place in an adjacent

3.2 Atomic jumps in diffusion

Fig. 3.40

89

Mechanism for cross-channel diffusion of adatom on Pt(110) proposed by Bassett and Webber. (a) Adatom in channel. (b) Vacancy forms in channel wall. (c) Adatom has moved into vacancy, lattice atom is now in channel site (after Bassett and Webber [40]).

channel and continued the diffusion. Bassett and Webber were unable to go beyond that, and warned that “further experimental and theoretical studies are required to clarify the nature of the inter-channel diffusion.” This followed rapidly. In 1979, Halicioglu [41] as well as Halicioglu and Pound [42] did calculations with Lennard-Jones potentials of the energetics of diffusion on channeled fcc planes. For Pt(110) they found cross-channel motion in which an adatom interacts with an atom from the channel wall to form a dumbbell-shaped pair. This could decompose in different ways, moving a lattice atom into an adjacent channel or else returning the adatom to its original channel. Shortly thereafter, De Lorenzi et al. [43,44] carried out detailed molecular dynamics simulations on fcc planes again bonded by Lennard-Jones potentials. On the (100) surface they found significant non-nearest-neighbor transitions at T = 0.49Tm, where Tm denotes the melting point. However, the most interesting results were for the (110) plane, which showed that atom motion occurred both along and across the channels, as in Fig. 3.41. For cross-channel motion they proposed an exchange of an adatom with a lattice atom, giving the model in Fig. 3.42, in which the transition state at the saddle point is made up of a lattice atom–adatom pair. From their simulations they deduced a lower activation energy for exchange than for inchannel motion. The work by Bassett and Webber [40] on Pt(110) stimulated much interest, so further simulations of cross-channel diffusion were also being carried out without, however, changing the overall picture very much. Mruzik and Pound [45] did more elaborate molecular dynamics simulations on several planes of a Lennard-Jones fcc crystal. Diffusion on (100) and (311) planes involved random jumping between nearest-neighbor positions. On (110), cross-channel motion again occurred by the exchange mechanism. Molecular dynamics estimates for gold and iridium adatoms were done by Garofalini and Halicioglu [46] on the Pt(110) surface with Lennard-Jones potentials at low and high temperatures. At low temperatures, both iridium and gold atoms diffused along the channels; at higher temperatures gold was found creating a dumbbell with a channel atom, but instead of exchanging rapidly returned to its original channel in which it continued to move. Overall movement was thus really one-dimensional as seen

90

Atomic events in surface diffusion

Fig. 3.41

Adatom trajectories in diffusion on fcc(110) plane at 0.4Tm. Transitions across the [1
10] channels are apparent. Based on Lennard-Jones potentials (after DeLorenzi et al. [44]).

previously by Bassett and Webber [40] in the FIM. With iridium, exchange occurred, a platinum atom ending up in an adjacent channel and continuing to move. A direct experimental demonstration of atomic exchange during cross-channel motion on fcc(110) was soon provided by Wrigley and Ehrlich [47] in 1980, who used the atom probe [48] to determine the nature of the atom making a cross-channel jump. They worked on the unreconstructed (110) plane of iridium. As a first step, the atom probe was calibrated by depositing adatoms on the surface maintained at 50 K, and then field desorbing them without heating the surface, while measuring the charge-to-mass ratio. The results for iridium as well as tungsten atoms are shown in Fig. 3.43. The next step was the crucial one. A tungsten atom was deposited on Ir(110), and the surface was heated until cross-channel motion occurred. The atom that appeared in the adjacent channel was then field evaporated and its charge-to-mass ratio determined. As indicated in Fig. 3.43, the atom measured after a cross-channel transition was iridium, a lattice atom. As a further test of the exchange mechanism, the adatom was removed from the surface, and the first lattice plane was field evaporated. After a cross-channel event, a tungsten atom was generally found in the surface layer. For planes on which crosschannel motion had not taken place, examination of the surface did not reveal a tungsten atom, as shown in Fig. 3.44. In 1982 Wrigley [49] examined the temperature dependence of cross-channel movement in self-diffusion on Ir(110), shown in Fig 3.45. He obtained an activation energy of 0.74±0.09 eV and a prefactor 1.4 (× 16±1)× 10−6cm2/sec. At that point it was not clear if such a low prefactor was characteristic of the exchange mechanism. That cross-channel motion involves atom exchange was now established, and at much the same time a new study showing cross-channel movement appeared. Tung and

3.2 Atomic jumps in diffusion

Fig. 3.42

91

Schematic of atom exchange process in self-diffusion on fcc(110) surface: (a) Atom in equilibrium position. (b) At the saddle point, atom pair sits as a dumbbell (indicated by ellipse) across [1
10] row of substrate atoms. (c1) – (c4) After diffusion, atoms distributed over allowed sites. See color plate section for color version of this figure.

Graham [16] in 1980 undertook quite a difficult task – they looked at self-diffusion on nickel surfaces. On the Ni(110) plane results depended very much on the pretreatment of the surface, as is clear from Fig. 3.46. Thermal treatment gave quite different diffusivities from what was found on a specimen field evaporated in hydrogen, but after both types of treatment the surfaces showed two-dimensional diffusion. The prefactors Do were very low on the thermally treated sample, 10−7 to 10−9 cm2/sec, with in-channel diffusion over a barrier of 0.23±0.04 eV and over 0.32±0.05 eV for cross-channel motion. With hydrogen exposure, in-channel movement took place over an activation energy of 0.30±0.06 eV and a more reasonable prefactor of 101 cm2/sec. For cross-channel events on a hydrogen fired surface the barrier was only 0.25±0.06 eV and a prefactor of 10−1 cm2/sec. In contrast, diffusion on Ni(311) and Ni(331) always occurred along

92

Atomic events in surface diffusion

Fig. 3.43

Distribution of atomic weights of individual atoms field evaporated from Ir(110) surface [47]. Top: After deposition of Ir atom. Bottom: After deposition of W atom. Center: After deposition of W atom, followed by heating and observing cross-channel diffusion event; material desorbed is iridium.

Fig. 3.44

Atomic weight distribution of material field evaporated from first Ir(110) layer [47]. Left: After cross-channel diffusion has occurred, a W atom is detected in the first substrate layer. Right: When no cross-channel motion was detected, only iridium is desorbed.

93

3.2 Atomic jumps in diffusion

10 × 10–16

Temperature T (K) 360 340

380

320

ED = 0.74± 0.09 eV Diffusivity D (cm2/sec)

Do = 1.4 (×16±1)× 10–6 cm2/sec

1

0.1

2.6

2.7

2.8

2.9

3.0

3.1

1000/T (K–1) Fig. 3.45

Early Arrhenius plot, based on FIM observations, for the diffusivity of Ir on Ir(110), involving cross-channel jumps (after Wrigley [49]).

Diffusivity D (cm2/sec)

180

160

Temperature T(K) 140 90

85

80

10–15

10–16

10–17

Ni/Ni(110) Thermally Annealed [001] Direction – [110] Direction

Ni/Ni(110) Hydrogen Etched [001] Direction – [110] Direction

10–18 60

Fig. 3.46

70

80 1/kT (eV–1)

130

140

Arrhenius plot for in- and cross-channel diffusion of Ni on Ni(110) plane. Left: Substrate after thermal treatment. Right: Nickel substrate has been hydrogen fired and field evaporated prior to experiments (after Tung and Graham [16]).

the channels. Tung [50] also examined self diffusion on Al(110), and found that at 154 K movement was two-dimensional with an activation energy of ~ 0.43 eV. It should be noted that these results for the (110) plane of Ni and Al are quantitatively uncertain. No new insights into cross-channel diffusion were offered, but they demonstrated quite unequivocally that cross-channel movement occurred on these surfaces.

94

Atomic events in surface diffusion

Fig. 3.47

Self-diffusion of Pt on Pt(110) surface observed in FIM. In-channel diffusion: ED = 0.72±0.07 eV, Do = 6×10−4 cm2/sec. Cross-channel diffusion: ED = 0.69±0.07 eV, Do= 3×10−4 cm2/sec (after Kellogg [51]).

Although it was now established that on fcc(110) planes surface diffusion could be two-dimensional, not much quantitative information was available about diffusion kinetics. Exchange was not observed on every fcc(110) plane; it did not occur, for instance, on Rh(110). In 1986 experiments resumed, however, and Kellogg [51] again looked at self-diffusion on Pt(110) and measured rates of in- and cross-channel motion at different temperatures, as shown in Fig. 3.47. For in-channel diffusion he found a barrier of 0.72±0.07 eV and a prefactor of 6 × 10−4 cm2/sec; for cross-channel movement the diffusion barrier was 0.69±0.07 eV, with a prefactor of 3 × 10−4 cm2/sec, both energies lower than first reported by Bassett and Webber [40]. Most important was the fact that the prefactors for exchange were similar to standard values. Further experimental tests were delayed for some years. Finally, in 1991, Chen and Tsong [52] carried out detailed measurements of self-diffusion on the (110) plane of iridium, and again observed cross-channel motion. In addition to the usual observations, they also quantitatively determined the distribution of displacements, shown in Fig. 3.48. Here x indicates the direction of in-channel motion and y cross-channel motion. From the temperature dependence of the mean-square displacements they arrived at the Arrhenius plot in Fig. 3.49, giving a lower activation energy of 0.71±0.02 eV for cross-channel movement than for movement along the channels; for this the diffusion barrier was 0.80 ±0.04 eV. The prefactors were 6 × 10−2.0±1.8 cm2/sec for cross-channel jumps and 4 × 10−3.0±0.8 cm2/sec for in-channel movement. What is of particular interest in this study is the distribution of displacements. Apart from the number of atoms at the starting position, with coordinates (0,0), the largest number was found at position (1,1), resulting from a cross-channel transition in the directions with 80% of the total. Atoms very rarely were seen in the (1,0) position, as a result of movements in the directions. From the dumbbell saddle configuration arrived at by all previous theoretical studies, we expect the same number of

95

3.2 Atomic jumps in diffusion

240.5 K, 20 sec 460 HP 0.243, 0.211

% 70

244.0 K, 20 sec 974 HP 0.441, 0.390

% 50

60 10 ⏐Δx⏐

0

0

1

2

⏐Δx⏐

0

3

0

1

2

3

1

1

2

2

⏐Δy⏐

⏐Δy⏐ %

%

248.0 K, 20 sec 316 HP 0.703, 0.649

50

⏐Δx⏐

0

0

1

2

0

3

0

1

2

⏐Δx⏐ 3

1

1

2

2

⏐Δy⏐ Fig. 3.48

260.5 K, 2 sec 449 HP 0.383, 0.303

50

⏐Δy⏐

Distribution of displacements in diffusion of Ir on Ir(110) observed in FIM at different temperatures. Transitions in direction of moving atom are favored, contrary to theoretical predictions. HP = heating period; last two numbers are mean-square displacement along x and y directions (after Chen and Tsong [52]). –1 Cross-Channel ED = 0.71 ± 0.02 eV Do = 6 × 10–2.0 ± 1.8 cm2/sec

–2

In N

–3 –4 –5

Along-Channel ED = 0.80 ± 0.04 eV Do = 4 × 10–3.0 ± 0.8 cm2/sec Ir on Ir (110)

–6 3.8

Fig. 3.49

3.9

4.0 1000/T (1/K)

4.1

4.2

Self-diffusion of Ir on Ir(110) plane examined in FIM. Both in-channel and cross-channel motion is observed (after Chen and Tsong [52]).

96

Atomic events in surface diffusion

transitions at (1,1) and (1,0), as the intermediate has the same probability of decomposing up or down in the neighboring channel. The results of these experiments were not accidental, however. Soon thereafter another examination was done by Kellogg [53] on platinum atoms on the (110) plane of nickel. What is worth noting is that the nickel surface was not cleaned in hydrogen; instead a combination of thermal cleaning, sputtering and evaporation was used. In the field ion microscope, Kellogg observed crosschannel movement of platinum atoms, and was able to recognize the identity of adatoms during atom exchange without resorting to the Atom Probe. The platinum adatom was found to field evaporate at the same voltage as the nickel substrate underneath did; nickel adatoms, on the other hand, were removed at 75–85% of the voltage for evaporating the substrate, making it easy to distinguish between the two species. The size of the image spot also turned out different: large for Pt, small for Ni. The image spot, seen after diffusion in Fig. 3.50b is small, and the adatom was field evaporated at a relatively low voltage, suggesting that the adatom on the surface after heating was nickel, not platinum. Additionally after removing the outermost layer, as in Fig. 3.50c, a platinum atom was left behind, establishing that atom exchange had indeed taken place. Also interesting were the observations of the location of the atom after cross-channel movement. Here the diagram in Fig. 3.2a is useful. Kellogg found that in 10 out of 16 observations the atom moved in the direction. That is, 62.5 % of the transitions were diagonally forward; the atom was never observed to translate in the direction. In the rest of the

Fig. 3.50

Neon field ion images showing exchange of Pt adatom with Ni from Ni(110) surface. (a) Single Pt adatom on Ni(110) plane. (b) After one minute at 112 K, adatom has changed to Ni, judged from lower desorption field. (c) Adatom has been removed by field evaporation. (d) After removal of one layer of nickel, Pt adatom is again apparent (after Kellogg [53]).

3.2 Atomic jumps in diffusion

97

observations atoms ended up in the original row. Kellogg associated this finding with the channel wall maintaining its cohesion with the adatom, an idea that was not confirmed, however, in earlier molecular dynamics studies. This result is very similar to the findings of Chen and Tsong [52]. Although these authors give no indication of being aware of previous theoretical studies, their results demonstrate that the picture of a dumbbell-like dimer intermediate, previously the unanimous result of theoretical endeavors, does not describe the real situation. More work is clearly needed to explain the mechanism of cross-channel transitions. Another year thereafter, Chen et al. [54] looked at Re diffusion on Ir(110), and were able to distinguish Ir from Re adatoms in terms of the shapes of the image spots: the image of Re was circular while Ir was elongated. They evaporated a Re atom onto the cold iridium, as in Fig. 3.51b, and then heated to ~ 256 K, moving the image spot one

Fig. 3.51

Exchange between Re adatom and Ir(110) substrate illustrated in field ion images. In (b), Re atom has been deposited on the clean Ir(110) surface in (a), giving a round image spot, a schematic of which is in (e). After one minute at 255 K, oblong atom, Ir, is found in (c), also illustrated in (f). Substituted Re atom is revealed on partial field evaporation, in (d) (after Chen et al. [54]).

98

Atomic events in surface diffusion

channel over. The image shape changed, as is clear in Fig. 3.51c. This came about through an atomic replacement of the Ir. After removing the outer layer of the substrate by field evaporation, they found in Fig. 3.51d a Re atom sitting on the surface – exchange had taken place with the iridium substrate; a schematic of the exchange is given in Fig. 3.51e–f. For proof of cross-channel diffusion it is not necessary to rely only on FIM observations. In 2002, Pedemonte et al. [55] did helium atom scattering experiments on Ag(110) at 750 K and higher. They found indications of in- as well as cross-channel movement, but the energetics of diffusion were not probed. Furthermore, beginning in the 1990s, much calculational work was started, usually resorting to semi-empirical potentials, characterizing diffusion on metal surfaces. These efforts are reviewed in Chapters 4 and 5. Here we just note that such work has been reported for (110) planes of aluminum [24,56–58], nickel [24,59–61], copper [24,60–66], palladium [24,60,61,67], silver [24,61,63,65,67,68], iridium [61,69,70], platinum [24,60,61,71,72], gold [24,60,61,65], and lead [73]. In most studies the activation energy for diffusion along the channels proved to be lower than across the channels. This despite the fact that in experiments on Al, Ni, Ir, and Pt the opposite was established. From all the work done so far it is clear that cross-channel diffusion occurs on some fcc(110) surfaces by exchange of the adatom with an atom from the lattice. On other fcc(110) surfaces, however, such as Rh(110), cross-channel transitions do not occur. Right now it is not clear why one particular system undergoes exchange while another one does not, and exactly how exchange occurs is not yet established. Nevertheless, atom exchange is an established fact. It should be noted that the relative energetics of crossversus in-channel diffusion vary from one material to the next, but the prefactors appear normal and of comparable magnitude. A new mechanism of surface diffusion beyond simple atom hops has been established.

3.2.2

Diffusion mechanisms: atom exchange on fcc(100) planes While work on atom exchange in diffusion on fcc(110) surfaces was still being actively pursued, DeLorenzi and Jacuzzi [74] in 1985 were continuing their efforts to find out more about diffusion, but this time on bcc surfaces. This, of course, required a new interaction potential, and for this they picked one developed by Price [75,76] to describe sodium, and examined atom movement on several planes. The one of primary interest here is the (100) plane. In diffusion on this surface an atom is expected to move from a normal site, located at the center of four substrate atoms, to an adjacent equilibrium site, in a direction at right angles to the boundary for the four lattice sites. What was found in their molecular dynamics simulations was quite different, as shown in Fig. 3.52: short, straight arrows indicate normal jumps between nearestneighbor sites, curved arrows emphasize exchange events in which an adatom replaces a lattice atom. The consequence is a diagonal transition quite different from expectations. In addition to the normal nearest-neighbor transitions, the adatom was found to exchange with an atom from the substrate, which then became an adatom at a site located diagonally from the starting position. The exchange event found here is

99

3.2 Atomic jumps in diffusion

ex j

ex j

ex

Fig. 3.52

Atom migration on bcc (100) surface, shown in molecular dynamics simulations modeled by Price potential [75,76]. Straight arrows j indicate nearest-neighbor jumps, curved arrows ex show adatom exchange with substrate (after DeLorenzi and Jacucci [74]).

Fig. 3.53

Schematic showing exchange between adatom and substrate atom on (100) plane: (a) Adatom at equilibrium site. (b) Adatom and lattice atom in transition state. (c) Adatom incorporated into lattice; atom from lattice has turned into adatom. See color plate section for color version of this figure.

modeled in Fig. 3.53. The original adatom moves into the outermost layer of the lattice, and the displaced lattice atom takes over on the surface. Quoting from the work of DeLorenzi and Jacuzzi, they found that “In addition to conventional nearest-neighbor jumps between surface sites, the adatom undergoes migration events reminiscent of exchange processes of the intersticialcy in the bulk. In these events, atom A belonging to the surface layer is replaced by atom B originally constituting the adatom. As a result, atom A ends up as an adatom located at a site displaced from the one originally occupied by the point defect.” “This is the first observation, in simulations or real experiments, of the occurrence of exchange events between adatom and a substitutional atom as a quantitatively important process contributing to atomic diffusion on isotropic crystal surfaces.” The stage was set now to explore whether diffusion by atom exchange would be important in a real two-dimensional system.

100

Atomic events in surface diffusion

Fig. 3.54

Map of sites on Pt(100) at which Pt adatom was found after diffusion at 175 K. A c(2×2) net is formed, indicative of transitions by atom replacement (after Kellogg et al. [80]).

–0.5

ED = 0.84 ± 0.05 eV Do = 6 .26 (× 11± 1)× 10–2 cm2/sec

In[/4t]

–1.0 –1.5 –2.0 –2.5 3.7

3.8

3.9

1000/T (K–1) Fig. 3.55

Arrhenius plot for diffusion of Ir adatom on Ir(100) observed in field ion microscope (after Chen and Tsong. [78]).

Nothing was done for several years to explore such exchange experimentally, but finally, five years later, Kellogg and Feibelman [77] examined Pt(100), and Chen and Tsong [78] more extensively studied diffusion on Ir(100). They looked at the displacements executed in diffusion. In normal jumps, atoms leap between nearest-neighbor sites. This is not what was observed, however. Instead, atoms were found to move diagonally, creating a c(2 × 2) net of sites as shown in Fig. 3.54. Kellogg and Feibelman estimated an activation energy of 0.47 eV for diffusion from the mean-square displacement at 175 K, assuming a prefactor of 1 × 10−3 cm2/sec. On Ir(001), Chen and Tsong actually made measurements at different temperatures to arrive at an activation energy for diffusion of 0.84±0.05 eV and a prefactor of 6.26(×11±1)×10−2 cm2/sec, all from the Arrhenius plot in Fig. 3.55. On both surfaces diffusion occurred as envisioned earlier by DeLorenzi and Jacuzzi [74], through exchange between the adatom and an atom from the substrate. The important end result of these measurements was the first experimental realization of atom exchange on a homogeneous two-dimensional surface.

101

3.2 Atomic jumps in diffusion

In[/t] (cm2/sec )

–32 Pt on Pt(100)

–34 –36 –38 –40 –42 5.0

ED = 0.47 ± 0.01 eV Do = 1.3 × 10–3 cm2/sec 5.2

5.4

5.6

5.8

6.0

1000/T (K–1) Fig. 3.56

Dependence of Pt atom diffusivity on Pt(100), observed in FIM, upon 1/T (after Kellogg [79]).

Fig. 3.57

Binding sites for Pd on Pt(100) after diffusion at 265 K, indicating ordinary hopping (after Kellogg et al. [80]).

There now followed a period of high activity studying (100) planes. Kellogg [79] further looked at diffusion on the (100) plane of platinum in 1991. He arrived at an activation energy for diffusion of 0.47±0.01 eV and a prefactor Do = 1.3(×10±1)× 10−3 cm2/sec from the Arrhenius plot of the rate in Fig. 3.56. The most important result here was to confirm that in self-diffusion on Pt(100), atoms describe a c(2 × 2) pattern of sites, characteristic of motion by atom exchange. For other atoms studied by Kellogg et al. [80] on Pt(100) the experimental situation was not as good. Detailed measurements at different temperatures were not possible, but for diffusion of palladium an activation energy of 0.70±0.01 eV was deduced assuming a prefactor of 10−3 cm2/sec. For nickel as well as platinum atoms, indications were that diffusion occurred by exchange, but for palladium ordinary jumps were involved, as

102

Atomic events in surface diffusion

Fig. 3.58

Interactions of Re atom with Ir(100) substrate observed in FIM. (a) Clean Ir(100) surface. (b) Re atom deposited on Ir surface. (c) Re-Ir dimer forms after heating to 230 K. (d) Dimer reorients after heating to 265 K. (e)–(f) Schematics of atom transitions (after Tsong and Chen [81]).

indicated by the (1 × 1) site map in Fig. 3.57. Note the much higher activation energy for diffusion of palladium compared to platinum, which is undoubtedly due to the difference in diffusion mechanisms. In 1992 Chen et al. [54,81] examined the behavior of rhenium on various iridium surfaces. They found that when a rhenium adatom, deposited on Ir(100), is heated above ~ 220 K as in Fig. 3.58, an iridium atom from the lattice is displaced to the surface and forms a dimer above the vacancy. Created in this way the dimer changes its orientation during annealing at 265 K. Eventually, further heating above 280 K brings about dissociation of the dimer, with the rhenium atom entering the lattice, while the iridium atom diffuses over the surface by atom exchange, a very direct demonstration of what happens in the exchange process. Kellogg [82] also looked at the behavior of platinum atoms on the Ni(100) surface. He was able to distinguish nickel from platinum atoms by the electric field required for field desorption: a nickel atom deposited on the plane was evaporated from the surface below the best voltage for imaging in the FIM at 77 K, whereas platinum atoms were

3.2 Atomic jumps in diffusion

Fig. 3.59

103

Neon field ion images of Ni(100) undergoing replacement with Pt adatom. (a) Single Pt atom deposited on (100) plane at 77 K. (b) On heating to 250 K, atom disappears. (c) Surface after partial field evaporation. (d) On complete field evaporation of plane, presence of Pt in the layer is detected (after Kellogg [82]).

maintained on the surface, and field desorbed just below evaporation of the substrate. In an experiment, a platinum atom was evaporated onto the surface at 77 K. After heating to 250 K the platinum atom disappeared, but after field evaporating the top nickel layer a platinum atom was again found there, as illustrated in Fig. 3.59, indicating an exchange event. Friedl et al. [83] examined self-diffusion on Ir(100) and again found a c(2 × 2) net for atom displacements. Instead of atom exchange they proposed a new idea – that the surface reconstructed, creating a c(2 × 2) sub-lattice for a diffusing atom, but this explanation did not survive closer examination. The same map of adsorption places should be observed for both Ir and Rh atoms if they move on a c(2 × 2) reconstructed sublattice. Fu and Tsong [84] tested this in 1999 and again found a c(2 × 2) net in selfdiffusion on Ir(100). Rh atoms, however, gave a (1 × 1) net in diffusion on Ir(100), characteristic of movement by hopping. It is important to recognize that quite a number of calculations have been carried out for diffusion on (100) planes. For Al(100) more than ten have been reported [24,56,58,85–92], with half giving atom exchange as the low energy process. Out of nineteen estimates for Ni(100) [24,92–104] only eight considered exchange as an option and fewer than half of these suggested an exchange process. Much more has been done on copper [24,62,86,89,92–94,96,98–102,104–125], but only half the studies considered exchange as an option and only three studies gave atom exchange as the low energy

104

Atomic events in surface diffusion

1.5 1.4

Hopping Exchange

1.3

ED (eV)

1.2 1.1 1.0 0.9 0.8 0.7 2

3

4

5

6

N Fig. 3.60

Dependence of activation energy for diffusion of Ni on Ni(100) plane upon the size of the unit cell (N×N) used in the ab initio calculations. Barrier to atom hopping is essentially constant; for adatom exchange it decreases strongly with size of cell (after Chang et al. [97]).

event. For Pd(100) [24,60,92–95,98–100,103,104] half indicated hopping was the likely diffusion process, but increasing the size of the cell used in the calculations started to favor exchange [94,98]. For Ag(100) [24,60,92–96,98–104,106,126–131], one effort considered atom exchange and indicated it as the favored process. The rest arrived at hopping as the leading mechanism. Relatively little calculational work has been done on Ir(100) [70,127], Pt(100) [24,132,133], and Au(100) [24,95,134,135]. However, all these gave atom exchange as the preferred mechanism of diffusion. One interesting fact was noted: in the theoretical studies, exchange depended on the size of the cell used in the simulations, as is apparent from Fig. 3.60. The theoretical calculations do not provide a clear indication of what is going on at (100) planes, but from the experiments it is evident that atom exchange is an important process in surface diffusion: in self-diffusion on Ir(100), Pt(100), Ni(100) and probably also on Au(100), diffusion occurs through exchange between the adatom and an atom from the substrate. It is still not certain why one atom diffuses by exchange and another by hopping. Feibelman [136] in 1990 first pointed to the covalent bonding in Al as stabilizing the transition state to atom exchange. Diffusion by atom exchange was correlated to the relaxation of atoms around the adatom by Kellogg et al. [80]. Tensile surface stress was believed to play a significant role in diffusion by atom exchange on fcc(100) planes by Yu and Scheffler [126]. However, detailed density-functional calculations for the (100) planes of Rh, Ir, Pd, and Pt by Feibelman and Stumpf [127] found no such clear relation. They felt that Kellogg’s proposal of substrate atom relaxation made sense. One thing is clear – predictions from experiments about diffusion by atom exchange are difficult.

105

3.2 Atomic jumps in diffusion

3.2.3

Diffusion mechanisms: multiple exchange The processes considered so far have all occurred through single events; however, this might not always be the case at elevated temperatures. Under such conditions more complicated mechanisms may start to play a non-negligible role. This has so far not been explored in experiments, which are difficult to do, but molecular dynamics simulations are feasible and have revealed interesting phenomena. The first of these was carried out by Black and Tian [105], who did molecular dynamics simulations of copper atoms on Cu(100), with interactions based on the embedded atom method. Their work was done for different diffusion temperatures. At 600 K they found atom exchange processes, but most interesting were simulations at 900 K. “A surface atom entered the substrate, and an atom not adjacent to the entering atom popped out of the substrate. The strain introduced by the entering atom then can be thought of as propagating along a row of substrate copper atoms.” Eventually an atom emerged from the substrate crystal at some distance from the original entry point and relieved the strain; a schematic is shown in Fig. 3.61. This appears to have been the first observation of multiple exchange events. A year later, Cohen [106] did molecular dynamics simulations on fcc(100) surfaces, again using embedded atom potentials. With an aluminum atom on the Al(100) plane at 500 K, she observed that an adatom embedded in the surface of the lattice, and “traveled two binding sites down the fcc(100) row and over one site.” The adatom moved on the surface like “a knight in chess.” Similar movement was also found for Ag, Au, Cu, Pd, Pt, and Ni, but the barrier for this new mechanism was much higher than for diffusion by the usual atom exchange, as is clear from the Arrhenius plot in Fig. 3.62. Cohen estimated that except for Ni(100), this new exchange process would be observable at temperatures

Fig. 3.61

t = 1.5 psec

t = 1.6 psec

t = 1.7 psec

t = 1.8 psec

t = 1.9 psec

t = 2.0 psec

Molecular dynamics simulation of multiple atom exchange for Cu on Cu(100) at 900 K. Black adatom moves into substrate, bringing about eventual emergence of substrate atom at a distance from original point of entry (after Black and Tian [105]).

106

Atomic events in surface diffusion

25 In (kh) (kh in #/sec)

Exchange

20

Hop

New 15

10

15

20

25

10 000/T (1/K) Fig. 3.62

Arrhenius plot for three different mechanisms of diffusion for Ag on Ag(110) derived in molecular dynamics simulations. New refers to multiple displacement exchange process, which has a higher activation energy and therefore only contributes at elevated temperatures (after Cohen [106]).

Fig. 3.63

Schematics of movements of Ag adatom on Ag(110) plane obtained in molecular dynamics simulations with RGL potentials at 600 K. Top: jump-exchange and exchange–jump events. Bottom: jump–exchange–jump. Dark grey shows atom in initial position, light grey indicates substrate atom participating in exchange (after Ferrando [137]).

above half the melting point. For nickel, temperatures would have to be even higher, but at high temperatures this multiple process certainly becomes competitive. A new type of jump process was uncovered by Ferrando [137] in 1996 who looked at molecular dynamics of Ag adatoms diffusing on Ag(110) using RGL potentials. In the vicinity of 600 K, that is at a fairly high temperature, he found the events illustrated in Fig. 3.63: an atom moving in a one-dimensional channel would after a jump carry out an exchange with an atom of the substrate, a jump–exchange event. Also observed in the simulations was the reverse process, exchange followed by a jump, as well as

3.2 Atomic jumps in diffusion

Fig. 3.64

107

Molecular dynamics simulation with RGL interactions of self-diffusion on Cu(100) at 950 K. Adatom squeezes into the surface layer, and eventually a surface atom pops out at a considerable distance from the first entry (after Evangelakis and Papanicolaou [110]).

jump–exchange–jump events, also illustrated in Fig. 3.63. Correlated movement was also observed on the Au(110) surface [65]. Other high temperature processes were pursued at greater length by Evangelakis and Papanicolaou [110] in 1996. They again did molecular dynamics simulations for a copper atom deposited on the Cu(100) surface, with atoms interacting via RGL type potentials. The simulations were carried out at elevated temperatures, above 700 K, and double-hop transitions were observed. At higher temperatures, they also found exchange processes in which the adatom entered the lattice and a lattice atom emerged nearby. A multiple exchange event is illustrated in Fig. 3.64. The adatom enters the surface layer and pushes aside a row of atoms, the last of which, four spacings away, finally emerges and transforms into an adatom. Evangelakis and Papanicolaou were able to follow these and similar processes at different temperatures, and obtained the Arrhenius plots in Fig. 3.65, which yielded the following diffusion characteristics: single jumps occurred over a barrier of 0.43±0.02 eV with a prefactor of 3.4 × 10−3±0.2 cm2/sec, for double jumps the values were 0.71±0.05 eV and 38 × 10−3±0.5 cm2/sec. For simple atom exchange the barrier was 0.70±0.04 eV, and the prefactor 42 × 10−3±0.3 cm2/sec, for double exchange 0.70±0.06 eV and 45 × 10−3±0.6 cm2/sec, for triple exchange 0.82±0.08 eV and 104 × 10−3±1.0 cm2/sec, and finally for quadruple exchange 0.75±0.12 eV and 86 × 10−3±0.9 cm2/sec. What is especially interesting here is that for multiple processes the activation energies as well as prefactors are all roughly comparable, with both substantially higher than for single atom jumps, so that multiple processes only occur at elevated temperatures. Calculations for self-diffusion on Pt(100) were carried out by Zhuang and Liu [133], who relied on EAM potentials [138–140]. They made estimates for diffusion by the

108

Atomic events in surface diffusion

Frequency (sec–1)

1011

1010

109

10

11

12

13

14

15

16

17

1/kT (eV–1) Fig. 3.65

Arrhenius plot for frequency of jumps in self-diffusion on Cu(100). Filled squares – single jumps; open squares – double jumps. Filled circles – simple exchange; filled and open triangles – double and triple exchange. Rhombic squares – quadruple exchange (after Evangelakis and Papanicolaou [110]).

Fig. 3.66

Model of ad-dimer diffusion mechanism on fcc (100) plane. Adatom penetrates into the lattice and dislodges two lattice atoms to form an ad-trimer state. When this decomposes, the third atom is left on the surface as an adatom (after Zhuang and Liu [133]).

standard exchange process, illustrated in Fig. 3.53, in which an adatom penetrates into the lattice and displaces a lattice atom to the surface. Zhuang and Liu also envisioned a more elaborate process, shown in Fig. 3.66. The adatom now enters the lattice, and displaces two adjacent lattice atoms to the surface to form an ad-trimer state. The atom in the center finally returns to the lattice, leaving the third one on the surface as an adatom in this ad-dimer diffusion. The end result is identical to what would transpire in a sequence of two standard exchanges, but the barrier to the new process proved to be somewhat lower, 0.26 eV, than for the conventional exchange, which required 0.28 eV. There are as yet no experiments demonstrating this ad-dimer mechanism, but it is something that should definitely be probed. Surprising results were obtained by Haftel and Rosen [141] in simulations using SEAM [138] in a molecular dynamics study to observe Au atoms condensing on Ag(111). They used constrained molecular statics, in which they moved the diffusing atom from the initial to the final step in small increments, relaxing all other atoms in every step. Their findings for the diffusion parameters on this close-packed surface are summarized in Table 3.2. Interesting here are the double exchange events occurring on the

109

3.2 Atomic jumps in diffusion

Table 3.2 Diffusion barriers and prefactors for Au on Ag(111) obtained in SEAM simulations [141]. Diffusion Type Hopping Exchange

Double Exchange Adatom catalyzed double exchange

Fig. 3.67

On terrace Over B step On terrace Over A step Over B step On terrace On terrace

ED (eV)

νD ð1012 sec
1 Þ

0.102 0.716 0.421–0.805 0.051–0.067 0.047 0.212 0.047–0.083

1.19

2.09 2.1

Schematic of exchange on Cu(111) of Cu adatom, in black, with lattice atom, pushing lattice atoms along until one moves to the surface at some distance from original entry of adatom. Events at high temperature and tensile stress found in EAM calculations (after Wang et al. [144]).

terraces as well as adatom catalyzed double exchange, which seems to be the lowest energy process. Haftel and Rosen claim that the ballistic energy of the colliding adatom, which they estimate as 0.11 eV, was able to catalyze double exchange, an effect rather unlikely in light of the studies presented in Section 3.4. The very detailed previous examination of surface processes by Evangelakis and Papanicolaou was rediscovered some years later by Xiao et al. [142,143]. Working with embedded atom potentials and the nudged elastic band technique to do simulations on Cu(100) and Pt(100), they observed multiple atom exchange in strained layers and referred to these events as crowdions. An atom emerging from this crowdion can of course contribute to the diffusion process. Shortly thereafter, Wang et al. [144] also published EAM calculations of self-diffusion on Cu(111). At high temperatures,  800 K, and tensile strains 2% or larger, diffusion by atom exchange became more probable even on this closely packed plane, but still with hopping as the leading mechanism. Especially interesting was the observation, shown in Fig. 3.67, of atom exchange, with a lattice atom being pushed out to the surface at some distance from the original exchange event. This complicated mechanism is only observed at high temperatures or under conditions of large strain. Action-derived molecular dynamics [145] estimates based on RGL interactions have also been made by Kim et al. [146] for a multiple exchange process on Cu(100). A Cu adatom penetrates partly into the first lattice layer, eventually causing a lattice atom to

110

Atomic events in surface diffusion

–38.34

Energy (a.u.)

–38.35

–38.36

–38.37

–38.38

Multiple exchange

0

200

100

300

Step Index Fig. 3.68

Energy of multiple exchange process for Cu adatom on Cu(100). 1 a.u. = 27.211396 eV (after Kim et al. [146]).

emerge six spacings away from the point of entry. The energetics for this process, in Fig. 3.68, are interesting, showing six peaks of the same height, roughly 0.8 eV and large compared to the barrier for single atom jumps, but comparable to the result of Evangelakis and Papanicolaou [110]. It would be interesting to know the prefactor for this process.2 What is clear from these various simulations is that multiple exchange events or crowdions may under the right conditions contribute significantly to diffusion. It must be recognized, however, that so far these are just the product of simulations. What still needs to be done now is to verify all this experimentally, which will require tests at elevated temperatures, or on stressed systems.

3.2.4

Diffusion on strained surfaces Up to this point, atom jumps have been considered as occurring on ordinary, thermally equilibrated surfaces. What happens, however, when a surface is strained? How does that affect atom diffusion? That is an effect first considered by Brune et al. [147], who did STM experiments on silver diffusion on both unstrained and strained Ag(111) surfaces. A strained surface was obtained by depositing silver on a Pt(111) plane, a layer shown to be under 4.2% compressive strain [148]. To determine the diffusivity they then measured the saturation island density. The Arrhenius plots they found are shown in Fig. 3.69a for Ag(111) as well as Pt(111), and on one monolayer of silver on Pt(111). On Ag(111), the activation energy for diffusion was 0.091±0.010 eV, with a frequency prefactor of 2 × 1011±0.5 sec−1. On the silver monolayer on Pt(111) the diffusion barrier 2

Kim et al. [146] have also commented on long jumps, but are clearly not well informed on the state of such studies.

111

–1.5

Diffusion barrier ED (meV)

log10 [saturation density (ML)]

3.2 Atomic jumps in diffusion

(a) –2.0 –2.5 –3.0 –3.5 –4.0 –4.5 0.005

Fig. 3.69

0.01

0.015 1/T (K–1)

0.02

0.025

80 (b) 70 60 50 40 30 0.90 0.95 1.00 1.05 1.10 Lattice constant [aAg (EMT) = 4.075Å]

Diffusion studies for Ag atoms on strained and unstrained surfaces. (a) Arrhenius plots for Ag diffusivity, derived from saturation island density. Square dots: Ag on Pt(111). Triangular dots: Ag on silver monolayer on Pt(111). Round dots: Ag on Ag(111). (b) Barrier to silver self-diffusion on Ag(111) as a function of strain, calculated with EMT (after Brune et al. [147]).

diminished to 0.060±0.010 eV, with a frequency prefactor 1 × 109±0.6 sec−1, and for Ag on Pt(111) the activation energy was high, 0.157±0.010 eV, and the prefactor turned out to be 1 × 1013±0.4 sec−1. The low barrier for diffusion on the silver monolayer could have been caused by electronic interactions with the underlying platinum. To check this, Brune et al. carried out EMT calculations for diffusion as a function of the lattice strain, shown in Fig. 3.69b. As the strain changed, the barrier to diffusion changed significantly from a reduction of the barrier for compressive strain to an increase when the strain is tensile. High tensile strains led to a maximum in the barrier followed by a strong decrease. The authors also showed that changes in the diffusion were mainly caused by changes in the binding energy at the equilibrium sites. Apart from the detailed effects, it is clear from these estimates that strain is an important parameter in surface diffusion,3 with compression smoothing the surface structure. Two years later, in 1997, Ratsch et al. [149] did density-functional estimates aimed specifically at the studies by Brune et al. [147]. For Ag diffusion on Pt(111), Ratsch et al. found a diffusion barrier of 0.150 eV, for Ag on Ag(111) the barrier was 0.081 eV, and for Ag on a monolayer of silver on Pt(111) the barrier was reduced to 0.063 eV, results in good agreement with the values from experiments. For silver self-diffusion on Ag(111) the effect of surface strain was calculated and is compared in Fig. 3.70 with the EMT calculations of Brune et al. [147]; in the former the diffusion barrier is quite linear, unlike the curved results obtained in EMT. What brings about the change in the diffusion barrier is revealed in Fig. 3.70b, which shows the energy of an adatom in the transition state, that is at a bridge site, compared to a normal fcc site. The difference between the two increases as the strain rises, with the binding energy at the fcc site increasing more rapidly. This

3

This was verified by Mortensen et al. [28] who used density-functional theory to estimate the effect of strain on the self-diffusion barrier for fcc metals. The derivative of the barrier height with respect to strain, in units of meV/%, turned out as follows: Ni, 2.8; Pd, 3.5; Pt, 3.6; Cu, 3.6; Ag, 3.8; and Au, 2.8.

112

Atomic events in surface diffusion

–2.3 DFT-LDA EMT

100

Total Energy (eV)

Diffusion Barrier ED (meV)

120

80 60 40

bridge site fcc site

–2.4

–2.5

–2.6 (b)

(a) 20

0.95

1.00 Relative Lattice Constant

Fig. 3.70

1.05

–2.7

0.95

1.00 Relative Lattice Constant

1.05

Diffusion of Ag on Ag(111) as a function of strain. (a) Comparison of EMT results from Brune et al. [147] with DFT calculations. (b) Energy of Ag atom in transition state (bridge site) and in normal fcc site on Ag(111) plane, as affected by lattice strain (after Ratsch et al. [149]).

altered the corrugation of the potential energy surface, making it more pronounced in expansion and flatter during compression. The energy difference between fcc and hcp sites was also estimated and turned out to be 0.04 eV for an Ag atom on a Pt(111) surface but only 0.01 eV for Ag on one monolayer of Ag on Pt(111). At much the same time as the investigations of Brune et al. [147], Meyer et al. [150] looked at the growth of nickel films on Ru(0001) with the STM, and found an increase in the mobility of Ni atoms as the number of nickel layers on ruthenium increased. The diffusion barrier for Ni was estimated to diminish by 0.30 eV in going from one nickel monolayer to three Ni layers. Meyer et al. pointed out that “the reason for the changing Ni mobility with layer height is not clear.” However, based on what is known at this moment, it is likely that strain could also be responsible for these observations. In 1997, Yu and Scheffler [151] employed density-functional calculations to diffusion on fcc(100) surfaces. They evaluated the diffusion barriers for atom hopping as well as for undergoing exchange as a function of the relative lattice constant, that is the surface lattice constant compared to that of the normal bulk material. As is clear from their work in Fig. 3.71, atom hopping is the favored process for the unstrained Ag(100) surface or under compressive stress; however, exchange becomes the important event under high tensile stress. Yu and Scheffler indicated that “while the significant tensile surface stress of Au(100) pulls the dimer of the exchange transition state ‘into’ the surface, lowers the energy of the transition state, and enables exchange diffusion, the surface stress at Ag(100) is too weak to have a significant effect.” “Smaller lattice constants correspond to a reduced corrugation of the surface potential, thus diffusion energy barriers are reduced. In contrast, for a stretched surface the corrugation increases, and the adsorption energy at the four-fold coordinated site increases.” It is interesting to note that the barrier height on (100) is again linear in the relative lattice constant, even out to high strains. Diffusion on strained surfaces was discussed by Schroeder and Wolf [152], also in the same year, but using Lennard-Jones potentials throughout, which limits the applicability to real systems. The effect of strain on the barrier for diffusion on a simple cubic (100)

113

Diffusion barrier ED (eV)

3.2 Atomic jumps in diffusion

Hopping Exchange

1.0

0.5

0.0 0.95

Fig. 3.71

1.00 1.05 Relative Lattice Constant

1.10

Self-diffusion barrier of Ag on Ag(100), derived in density-functional calculations, in its dependence on strain (after Yu and Scheffler [151]).

Activation Energy

(a)

Energy

–0.8

–1.1

(b)

0.4

0.3

0.2 –0.10 –0.05

–1.2 0.0

1.0 Position

Fig. 3.72

0.5

0.00 Strain

0.05

0.10

Effect of strain on self-diffusion on a (100) plane. (a) Potential energy change in self-diffusion on a Lennard-Jones (100) plane for different strains. Middle curves: unstrained. Upper curves: tensile strain, εxx ¼ εyy ¼ 0:05. Lower curves: compressive strain, εxx ¼ εyy ¼
0:05. Filled symbols: global relaxation. Open symbols: frozen lattice. (b) Dependence of self-diffusion barrier on Lennard-Jones (100) plane upon strain (after Schroeder and Wolf [152]).

surface is shown in Fig. 3.72a, with the upper curve calculated for tensile strain εxx ¼ εyy ¼ 0:05, and the bottom one for a compressive strain of 0.05. The central curves give values for the normal system as a function of the x-coordinate of an adatom as it moves from one binding site to an adjacent one. The activation energy as a function of surface strain is plotted in Fig. 3.72b for diffusion on the (100) plane of a simple cubic crystal, increasing as strain becomes more tensile. According to Schroeder and Wolf, changes in the barrier height are mostly due to changes in the saddle point, rather than the changing in the strength of the binding energy. Much the same effects were found for fcc as well as bcc(100) planes. For an fcc(100) crystal, the binding energy weakly depended on the strain, so again changes in the barrier are mostly due to changes in the saddle point. However, the situation differs for a bcc(100) surface, where changes in the strength of

114

Atomic events in surface diffusion

binding caused by strain are significant. This is due to the presence, in the second lattice layer, of an atom just below the binding site of the adatom, which contributes to binding energy changes. More recently, Liu et al. [153] used glue potentials [154,155] to examine the influence pffiffiffi of local strain on the surface self-diffusion of Au on the (23  3)-Au(111) reconstructed surface. A schematic of this surface is given in Fig. 3.73a; B and D indicate fccand hcp-like regions, A, C, and E are transition regions. For self-diffusion on the unreconstructed Au(111) this method yielded a value of 0.7 eV, which the authors

Fig. 3.73

pffiffiffi pffiffiffi Diffusion on the reconstructed (23  3)-Au(111) surface. (a) Schematic of (23  3)-Au(111) surface. B and D denote fcc-and hcp-like regions, A, C, and E indicate transition regions. Preferred directions for diffusion shown by thick arrows. (b) Estimate of strain in different regions of the surface. (c) Self-diffusion barrier at different positions estimated using glue potential (after Liu et al. [153]).

115

3.2 Atomic jumps in diffusion

Diffusion coefficient (10–4 cm2/sec)

1.35 T = 650 K uniaxial strain biaxial strain

1.30 1.25 1.20 1.15 1.10 1.05 1.00 0.95 1.00

Fig. 3.74

1.01 1.02 1.03 Relative Lattice Constant

1.04

Diffusivity of Cu adatom on Cu(111) plane, calculated with EAM potentials in its dependence on strain (after Wang et al. [144]).

recognized as too high. Only general features can therefore be compared. The binding energy at fcc sites is higher than at hcp sites, and based on this observation the authors concluded that strain does not change the energy sequence. An estimate of the local strain at various locations is plotted in Fig. 3.73b and the calculated diffusion barrier is indicated in Fig. 3.73c. Here Ex
stands for the barrier in the direction f–g shown in Fig. 3.73a, and Ex is for the direction f–h. It is clear that diffusion is fastest in the transition regions A, C, and E, where the strain is high, so these regions create easy paths for adatom movement. A year later, in 2003, Wang et al. [144] performed molecular dynamics simulations for diffusion on Cu(111) using EAM potentials from Mishin et al. [31]. The strong effect of both uniaxial as well as biaxial strain on the diffusivity is clear in Fig. 3.74, with tensile strain diminishing the diffusivity. High values of the tensile strain also bring about diffusion via atom exchange, absent on a normal Cu(111) surface. In 2004 Xiao et al. [143] looked at adatom diffusion on a strained Pt(100) surface, investigating the influence of the final size of the computational cell on the energy barrier. As is clear from Fig. 3.75a, they found that the dependence of the energy for exchange on strain was cell-size dependent, and associated this with the displacement of atoms over longer distances during adatom exchange, as in Fig 3.75b. If such displacements are not EAM artifacts, then the size of the cell used can cause systematic overestimation of the energy for atom exchange. The change in hopping energy versus strain for differently sized cells is illustrated in Fig. 3.76a; for small cells the expected trend of changes with increasing strain is reversed compared to bigger cells. A comparison of the energy change for hopping and exchange with strain on a big 10 × 20 cell is presented in Fig. 3.76b, the barrier rising for atom hops and decreasing for exchange events. Although the studies of strain effects on surface diffusion have relied on a variety of different techniques, one thing is quite clear: strain exerts a big effect on diffusion kinetics, and is something to take advantage of in guiding diffusion. More experimental

116

Atomic events in surface diffusion

Energy Barrier ED (eV)

0.7 Pt cell 10 × 20 Pt cell 5 × 5

0.6 0.5 0.4 (a) 0.3 0.2 –0.04

Fig. 3.75

–0.02 0 0.02 biaxial tetragonal strain εxx = εyy

0.04

(b)

Effect of strain on the self-diffusion barrier on Pt(100) calculated with EAM. (a) Dependence of exchange energy on size of computational cell. (b) 10 × 20 unit cell used in computation, showing initial configuration at the top, and transition state for unstrained surface at the bottom (after Xiao et al. [143]). 0.5 Pt hopping 10 × 20 Pt exchange

0.5 0.4

5×5 6×6 7×7 8×8 9×9 10 × 10 10 × 20 10 × 30

0.3 (a) 0.2 –0.03 –0.02 –0.01

Fig. 3.76

0

0.01 0.02 0.03 0.04

εxx = εyy

Energy Barrier ED (eV)

Energy Barrier ED (eV)

0.6

0.4

0.3 (b) 0.2 –0.04

–0.02 0 0.02 Biaxial tetragonal strain

0.04

Diffusion of Pt atom on strained Pt(100) surface. (a) Diffusion barrier for atom hopping in its dependence on the computational cell size. (b) Effect of biaxial strain on barrier to hopping and also to exchange (after Xiao et al. [143]).

information will be useful in this area to start deriving information about absolute values of diffusion barriers, not just about trends.

3.3

Long jumps in surface diffusion

3.3.1

Theoretical studies Even in the 1970s the picture that in diffusion atoms jump randomly between nearestneighbor sites on the surface still prevailed. However, there gradually appeared theoretical efforts countering this view. In 1979, Tully et al. [156] made ghost particle simulations for diffusion on a (100) plane of a Lennard-Jones crystal at temperatures below the melting point Tm. As shown in Fig. 3.77, they realized that the mean jump length

117

3.3 Long jumps in surface diffusion

Correlated jump distance

4

3

2

1

0

0.2

0.4

0.6

0.8

Temperature T Fig. 3.77

Temperature dependence of the length of correlated jumps in diffusion of adatom on Lennard-Jones fcc(100) surface (after Tully et al. [156]).

increased from 1.5 spacings at a low temperature to 3.5 spacings at high values. At much the same time extensive simulations, again with Lennard-Jones potentials, were done by DeLorenzi et al. [43,44]. They examined several fcc crystal planes, but most interesting were their results for the (100). At T ~ 0.5Tm long jumps were observed, sometimes between sites three or more spacings apart, as is clear from their trajectories in Fig. 3.78. On the fcc(110) plane they found that diffusion was two-dimensional, occurring both along and across the channels, with exchange between the adatom and an atom from the lattice. Adatom behavior on the close-packed (111) plane at T ~ 0.44Tm proved to be quite different, as seen in Fig. 3.79. Rather than hopping between sites, the adatoms carried on close to free two-dimensional motion. The same year, Mruzik and Pound [45] also did molecular dynamics estimates of diffusion on an fcc Lennard-Jones crystal. On the (111) plane at T* = 0.1, where T* = kT/εLJ, and εLJ is the energy parameter in the Lennard-Jones potential, diffusion occurred essentially between nearest-neighbor positions, but at higher temperatures, at T* = 0.45, atom motion resembled free flight over the surface, as already shown in Fig. 3.79. The atoms spend little time at the binding sites and long jumps occur. The situation is different on the (311) plane, where atom motion is entirely along the channels on the surface, depicted in Fig. 3.2b. For the (110) plane, diffusion at T* = 0.45 occurred by exchange between the adatom and the lattice. The picture developed by Mruzik and Pound is essentially the same as that of DeLorenzi et al. [43,44], but more detailed. Lennard-Jones interactions were again used by Voter and Doll [157] in 1985 for treating transitions longer than nearest-neighbor spacings in transition-state theory calculations of diffusion on Rh(100). However, longer jumps were found only at quite high temperatures, around 1000 K. In the meantime, DeLorenzi and Jacuzzi [74] had continued their work, and in the same year published their estimates for bcc surfaces,

118

Atomic events in surface diffusion

Fig. 3.78

Adatom trajectories on Lennard-Jones (100) surface at T = 0.49Tm, revealing non-nearest-neighbor transitions (after DeLorenzi et al. [44]).

using a metallic potential originally intended for sodium [75,76]. All simulations were done at T ’ 0:4Tm . As already pointed out previously, they noted exchange events on the (211) plane, and more importantly, on the (100) face. Most interesting in the present context were the simulations on the (110) surface, the densest on the bcc lattice. There DeLorenzi and Jacuzzi observed single jumps, in Fig. 3.80a, but of greater interest, double jumps in Fig. 3.80b, and even more complicated transitions. The simulation effort was rounded out a few years later by the work of Sanders and DePristo [158] in 1992, who made molecular dynamics calculations on the fcc(111) surface, approximating atomic interactions by the corrected effective medium theory, but also by Morse and Lennard-Jones potentials. What they found was not surprising. At 800 K, the time an atom spent at a given site, was, as suggested in Fig. 3.81, very small, indicating that the atom did not fully equilibrate there. Ferrando et al. [159] in 1993 set about to solve the problem of a Brownian particle moving over a one-dimensional periodic potential in an external field. Long jumps are expected to appear when exchange of energy between the adatom and the lattice becomes poor and Ferrando et al. were able to calculate the probability of jumps longer than a nearest-neighbor spacing as a function of the dissipation Δ in traveling over the tops of the barriers, given by Δ¼

4‘0 ðmED Þ1=2 : πkT

(3:6)

3.3 Long jumps in surface diffusion

Fig. 3.79

119

Delocalized adatom trajectories on fcc(111) surface at 0.44Tm, obtained by simulations with metallic potential (after DeLorenzi et al. [44]).

Here  is the coefficient of friction between the diffusing atom and the lattice, and the probability of long jump decreases as the dissipation increases. This approach to the diffusion of a particle has been extended by Georgievskii and Pollak [160,161] and by Chen and Ying [162], who arrived at the probability of a jump of many nearest-neighbor spacings in its dependence upon the magnitude of the barrier. What turned out to be of interest is the small effect of temperature upon the length of the jumps. In 1994, Ellis and Toennies [163] made an interesting discovery. They carried out molecular dynamics simulations for the diffusion of Na on Cu(100), using Morse potentials, and arrived at the Arrhenius plot in Fig. 5.24, giving an activation energy of 0.051±0.006 eV and a frequency factor of (0.53±0.2) × 1012 sec−1. More importantly, they also discovered that for every nearest-neighbor transition of the Na atom, long jumps occurred more than 0.7 times at 200 K, and just a little less at 300 K. Somewhat later, Evangelakis and Papanicolaou [110] observed double jumps on Cu(100) at temperatures above 750 K, as was already mentioned in Section 3.2.3. In 1996, Ferrando [137] examined simulations of self diffusion on Ag(110) and observed that 90% of the atom jumps were single transitions at any temperature; the

120

Atomic events in surface diffusion

Fig. 3.80

Adatom trajectories on bcc (110) plane modeled with Price potential [75,76] during 200 psec at ~ 0.4Tm. (a) Single jump. (b) Double jump. (c) Complicated trajectories (after DeLorenzi and Jacucci [74]).

Fig. 3.81

Trajectory of Rh atom on Rh(111) plane at 800 K, obtained in molecular dynamics simulations with MD/MC-CEM (after Sanders and DePristo [158]).

3.3 Long jumps in surface diffusion

121

rest were mostly doubles. A few years later, Ferrando extended his study to diffusion on (110) surfaces of Au, Ag, as well as Cu [65]. Despite similarities in the energetics of diffusion on Au and Ag, and a slightly lower barrier for Cu, the behavior of long jumps proved to be quite different on the different surfaces. At 450 K, long jumps were absent on gold, there were 3% long jumps on silver, and 6% on copper. For copper, the fraction of long jumps increased to 15% at 600 K, but never got to this value on the Ag(110) surface. In the meantime Azzouz et al. [164] derived a master equation for atoms hopping on a crystal, and found that for atoms weakly interacting with the surface transitions beyond nearest-neighbor sites became more likely, in agreement with the work of Ellis and Toennies [163]. In 2004, Ferron et al. [165] carried out molecular dynamics simulations relying on EAM interactions. They saw long jumps as well as rebound (recrossing) jumps in self-diffusion on Cu(111) and claimed that at 500 K, 95% of jumps were correlated; at 100 K this decreased to 50%. They continued their simulations using the DYNAMO code and atomic interactions derived from EAM potentials [166]. A number of correlated jumps, including atomic long events as well as ballistic transitions were identified at temperatures ranging from 7% to 55% of the melting temperature of copper, with the likelihood of correlations increasing with increasing temperature. Also observed was a linear increase of the characteristic jump length with temperature. The average distance covered by adatoms increased faster than expected for a random walk between nearestneighbor sites. The influence of long jumps on island morphology on fcc(100) surfaces was studied by Beausoleil et al. [167] with Monte Carlo simulations based on bond-counting energetics. They considered only long jumps forward, ignoring rebound jumps, and found that due to long jumps island coalescence is slowed down, adatom density as well as island density increased, but the average size of islands is lowered. These simulations, although quite different from each other, have all shown that at elevated temperatures long jumps, that is jumps in which the adatom does not equilibrate at a nearest-neighbor site, instead spanning two, three, or longer nearest-neighbor distances, can be expected to take place at higher temperature. What these temperatures are, and what the type of jump is, does not emerge, as the interatomic forces were only roughly approximated. The emphasis therefore turns to experiments: how can long jumps in single atom diffusion be detected?

3.3.2

Distribution of displacements Long jumps cannot be seen directly in experiments, due to the lack of an experimental technique fast enough to probe jumps in real time and at the same time to have control of a single adatom. One possible way to distinguish experimentally between single and long jumps is measurement of the distribution of displacements. Usually information about  2 diffusivities is obtained from measurements of the displacement fluctuations, are related to the diffusivity D through the Einstein relation Δx . These  fluctuations (Eq. 1.36), Δx2 ¼ 2Dt. For diffusion via single jumps between nearest-neighbor sites the diffusivity D can be written from Eq. 1.18 as D ¼ G1 ‘20 , so that

122

Atomic events in surface diffusion

γ –4

Fig. 3.82

β –2

α

α 0

γ

β

x

2

4

Schematic of single, double, and triple transitions in one-dimensional symmetric atom motion.



Δx2 ¼ 2G1 t‘20 ;

(3:7)

where G1 is the rate of nearest-neighbor transitions and ‘0 is the nearest-neighbor jump length. Suppose now that diffusion is one-dimensional and involves not only single jumps but also jumps of arbitrary length ‘i , as illustrated in Fig. 3.82. Then 

X Δx2 ¼ 2t Gi ‘2i ;

(3:8)

i

where the summation extends over all types of jumps i and their rates Gi . The diffusivity is therefore given by D¼

X

Gi ‘2i :

(3:9)

i

It is evident that long jumps should affect the diffusivity, but this by itself does not allow a determination of what the rate of long jumps amounts to. In terms of transition-state theory, the diffusivity has been written as Eq. (1.21)       ΔSD ΔED ΔED 2 D ¼ ν‘ exp exp
¼ Do exp
. From a plot of the natural k kT kT log of the diffusivity D against 1/kT we can arrive at the activation energy, which henceforth will be written as just ED, and also the prefactor Do. The value of the prefactor will depend upon the frequency ν, the mean-square jump length ‘2 , and the entropy of activation ΔSD . These contributions to the prefactor Do are not easy to sort out, so that ‘2 is not directly accessible in this way. However, the displacements measured to get information about the diffusivity D can also provide further insights if a sufficient number of measurements are available. It is then possible to ascertain the distribution of displacements, shown in Fig. 3.83 for one-dimensional diffusion with a mean-square displacement of two. Expressions for the distribution have already been derived in Section 1.2.3. If, in addition to nearest-neighbor transitions there are also double jumps, spanning the distance between next-nearestneighbors, then px, the probability of being at a distance x from the origin, is given by 1 P (Eq. 1.104) px ¼ exp½
2ðα þ βÞt Ij ð2βtÞIx
2j ð2αtÞ, where α is the rate of single j¼
1

jumps, β the rate of double transitions, and Ii(z) is the modified Bessel function of order i and argument z. Suppose that in this migration the mean-square displacement is again

123

3.3 Long jumps in surface diffusion

5000

Single Jumps

Double/Single = 1 5000

4000

4000

3000

3000

pΔx N

pΔx N

= 2.0

2000 1000 0 –8 –6 –4 –2 0 2 4 Displacement Δx

Fig. 3.83

= 2.0

2000 1000

6

8

0 –8 –6 –4 –2 0 2 4 Displacement Δx

6

8

Effect of double jumps on the distribution of atomic displacements in diffusion with a mean-square displacement of two.

two, but double jumps occur at the same rate as singles. This distribution is also shown in Fig. 3.83 at the right, and is obviously quite different from what we have seen for single jumps, at the left. The number of zero displacements is much higher, and the adatom is less frequently seen at longer distances from the origin. By analyzing the distribution of displacements we obviously can gain information about the jump rates in diffusion. In practice there are of course some problems. Equation (1.104) gives the expression for the distribution of displacements on an infinite line of sites. In the field ion microscope, however, the individual planes are small, usually less than 60 Å in diameter, so that this equation is not really applicable. Furthermore, close to the edges of the plane, we have seen that two or three sites may have properties different from the interior and have to be excluded. To deduce jump rates from the measured distribution we therefore have to resort to Monte Carlo simulations. These are carried out on a plane of the same size, shape, and structure as the plane in the experiments. In the simulations the adatom is allowed to make all predicted kinds of jumps and jump rates are varied until the best agreement is found between simulated and experimental distribution. It should also be noted that here we have been concerned entirely with onedimensional diffusion, but this is not a real limitation. In diffusion on a two-dimensional surface, the distribution along x and y can usually be analyzed separately. There are still some corrections to be made, but these have already been discussed in Chapter 2.

3.3.3

Early experiments The first attempt to experimentally probe the distribution of displacements during diffusion in order to elicit information about jumps to other than nearest-neighbor sites was made in 1989 by Wang et al. [168] on a one-dimensional system: W(211). Distributions of displacements were measured in the FIM over a range of temperatures for Re, Mo, Ir, and Rh. In Fig. 3.84 are shown distributions for rhenium at 300 K for two different time intervals and for molybdenum at 262 K in Fig. 3.85. The best fit to these distributions was derived with a ratio of double-to-single jumps equal to zero, that is without any long jumps involved. However, it is clear that the continuous time

124

Atomic events in surface diffusion

800

β/α = 0 Experiment Gaussian

Total Observations 600 1112 400

200

0

600

Re on W(211) T = 300 K, 20 sec = 0.39 Total Observations 1120

β/α = 10–3 Experiment Gaussian

400

200

–4

–2

0

2

4

Displacement, Δx (Nearest-neighbor spacing)

Fig. 3.84

Observations

Observations

Re on W(211) 800 T = 300 K, 10 sec = 0.23

0

–4

–2

0

2

4

Displacement, Δx (Nearest-neighbor spacing)

Distribution of Re adatom displacements at 300 K on W(211), obtained from field ion observations [168]. Best fit to experiments obtained with an entirely negligible contribution from long jumps. 600

Number of Observations

500

400

Mo on W(211) T = 262 K, 10 sec = 0.84

β/α = O

Total Observations 971

Experiments Gaussian

300

200

100

0 –4 –2 0 2 4 Displacement, Δx (Nearest-neighbor spacing) Fig. 3.85

Distribution of Mo adatom displacements on W(211) at 262 K [168]. Only nearest-neighbor transitions are found.

description of diffusion outlined in the previous section correctly describes atomic behavior – atoms migrated between nearest-neighbor sites. Only for iridium and rhodium diffusion, in Fig. 3.86, were small contributions from double jumps found, but these were too small to be really significant. These early studies were important in gaining experience, but also in confirming the simplest picture of diffusion. The next significant step in the search for long jumps was taken by Senft [169–171]. She decided that long jumps were more likely to occur if energy exchanges between

125

3.3 Long jumps in surface diffusion

1000

600

Ir on W(211) T = 261 K, 5 sec = 1.00

β/α = 0.044

Total Observations 1208

β/α = 0

Experiments

400

200

Number of Observations

Number of Observations

800

Rh on W(211) T = 189 K, 10 sec 800 = 0.44 Total Observations 1388

Fig. 3.86

Experiments

β/α = 0

600

400

200

(a) 0

β/α = 0.044

(b)

–4 –2 0 2 4 Displacement, Δx (Nearest-neighbor spacing)

0

–4 –2 0 2 4 Displacement, Δx (Nearest-neighbor spacing)

Distribution of displacements for Ir and for Rh adatoms on W(211) plane, suggesting a tiny contribution of double jumps [168]. 700 Pd on W(211)

Number of Observations

600 500 400

T = 114 K, 40 sec = 0.45 951 Observations

Single jumps only Experiment β/α=0.014 ± 0.024

300 200 100 0

Fig. 3.87

–3 –2 –1 0 1 2 3 Displacement, Δx (Nearest-neighbor spacing)

Distribution of Pd adatom displacements on W(211) plane at 114 K, showing essentially single jumps only (after Senft [169]).

adatom and lattice were small, as then the atom might bounce after the first impact and continue farther on. She therefore picked palladium and nickel to study on W(211), as these metals have low barriers to diffusion, 0.314±0.006 eV for palladium and 0.46±0.01 eV for nickel, that should lead to long jumps. At 114 K, the distribution for displacements of palladium is shown in Fig. 3.87. Essentially only single jumps contribute to the distribution at this temperature; the ratio of double-to-single jumps amounted to only ~ 1%, too small to be significant. Measurements at 122 K revealed

126

Atomic events in surface diffusion

350 300

Observations

250

Pd on W(211) T = 133 K, 3 sec = 3.76 1164 Observations

Single jumps only Experiment β/α = 0.20 γ/α = 0.13

200 150 100 50 0

Fig. 3.88

–10

–8 –6 –4 –2 0 2 4 6 Displacement, Δx (Nearest-neighbor spacing)

8

Distribution of Pd atom displacements on W(211) at 133 K. Best fit with double/single jumps equal to 0.20, and triple/single of 0.13 (after Senft [171]).

500 Ni on W(211)

Observations

400 T = 179 K, 2 sec = 1.17 300

947 Observations

Single jumps only Experiment β/α = 0.058 ± 0.038

200

100

0

–4 –3 –2 –1 0 1 2 3 4 Displacement, Δx (Nearest-neighbor spacing) Fig. 3.89

Distribution of Ni adatom displacements on W(211) plane at 179 K. Best fit of observations with double/single jumps equal to 0.058 (after Senft [170]).

essentially the same situation, but everything changed dramatically at a slightly higher temperature of 133 K, as indicated by the distribution in Fig. 3.88. Now the experiments differed significantly from the distribution for only single jumps. The best fit was obtained for a ratio of double-to-single jumps of 0.20±0.06, and even triple jumps contributed, at a ratio of 0.13±0.04 to singles. This marked the first time that long jumps had been found in a real system. And it happened at quite a low temperature, at less than one twenty-fifth the melting point of tungsten. For the diffusion of nickel, Senft also found long jumps, as is clear from the diagram at 179 K in Fig. 3.89. In this system, the ratio of double-to-single jumps turned out to be 0.058±0.038. The behavior of tungsten self-diffusing on W(211), studied by Senft is quite different. As shown in

3.3 Long jumps in surface diffusion

127

500

Observations

400

= 1.24 β/α =γ /α = 0

Experiment Best Fit

300 200

100 0

–5 –4 –3 –2 –1 0 1 2 3 4 5 Displacement, Δx (Nearest-neighbor spacing) Fig. 3.90

Displacement distribution of W adatom on W(211) at 307 K. Best fit to field ion experiments derived with negligible contribution of β double or longer jumps (after Senft [170]).

Fig. 3.90 at 307 K, these experiments are best represented by a distribution with a quite negligible number of double jumps; only nearest-neighbor transitions mattered. However, Senft did not investigate higher temperatures for a tungsten adatom. What is especially interesting about the results for palladium is the very strong dependence of the frequency of long jumps on temperature. A drop of only eleven degrees, to 122 K, makes all transitions other than single jumps vanish. Shortly before this work, Ferrando et al. [159] worked out a theory for the random motion of an atom over a periodic potential in a thermal bath subject to frictional forces, and came up with a description for the probability of long jumps. However, in this theory the occurrence of long transitions has only a small dependence upon the temperature, and is therefore not in accord with Senft’s findings of a strong sensitivity to temperature, as if long jumps were characterized by a larger activation energy. With the existence of long jumps established, at least in one-dimensional diffusion, attention now turned to the detailed temperature dependence. Linderoth et al. [172] in 1997 studied this in detail for the diffusion on the reconstructed Pt(110)-(1 × 2) surface, a model of which, showing wide channels, is given in Fig. 3.91. Linear clusters and single atoms can easily be seen on the STM image of this surface in Fig. 2.9. Motion was examined with the STM after deposition of a sub-monolayer quantity of platinum. An example of the distribution of displacements found at 375 K is shown in Fig. 3.92, for which the ratio of double-to-single jumps was reported as 0.095. From measurements at temperatures over an ~ 60 K range they came up with the Arrhenius plot in Fig. 3.93, which gave them an activation energy of 0.81±0.01 eV with a prefactor ν0 ¼ 1010:70:2 sec
1 for single jumps, and a slightly higher barrier of 0.89±0.06 eV and roughly the same prefactor, 1010.9 ±0.8 sec−1, for what were presumed to be double jumps. This would represent the first estimate of the barrier to long jumps, but the situation changed rapidly. The next year, Montalenti and Ferrando [173] did molecular dynamics simulations on the (1 × 2) reconstructed Au(110) plane, modeled in terms of the RGL potential

128

Atomic events in surface diffusion

Fig. 3.91

Hard-sphere model of reconstructed fcc(110)-(1 × 2) plane.

Single only Experiment Single & Double

600

T = 375 K Δt = 2.3 sec 1620 Observations = 1.68

Observations

500 400 300 200 100 0 –4

–3

–2

–1

0

1

2

3

4

Displacement (lattice spacing) Fig. 3.92

Distribution of atomic displacements in self-diffusion on Pt(110)-(1 × 2) plane. Best fit obtained with ratio of “double” to single jumps of 0.095 (after Linderoth et al. [172]).

[174,175]. What they found is shown in Fig. 3.94. Jumps could take place in a straight line on the bottom of the channels; these could span one, two or three spacings. What they also observed was that atoms could carry out a metastable walk, in which an atom jumps up the (111) channel walls and continues diffusion there. The number of these metastable walks exceeded the number of long jumps at the bottom at all temperatures, from 450 K to 625 K, and ~ 20% of the transitions over two spacings occurred by metastable walks. Although Montalenti and Ferrando made no inferences concerning the results of Linderoth et al. [172], this happened the next year.

129

3.3 Long jumps in surface diffusion

Temperature T (K) 385 1

357

333

313

294 Single Double

E1 = 0.81 ± 0.01 eV

Rate (1/sec)

0.1

0.01

0.001 E2 = 0.89 ± 0.06 eV 0.0001 2.6

2.8

3.0

3.2

3.4

1000/T (1/K) Fig. 3.93

Arrhenius plot for single and double jump rates presumed to occur in self-diffusion on Pt(110)(1 × 2) (after Linderoth et al. [172]).

Lorensen et al. [176] in 1999 carried out density-functional calculations of the energy barriers in diffusion on the reconstructed Pt(110)-(1 × 2). They again found that not only could atoms jump in a straight line along the bottom of the channels, but, as shown in Fig. 3.95, they could also carry out metastable walks on the (111) facets of the channel sides, as had been reported earlier by Montalenti and Ferrando [173], with energetics indicated in Fig. 3.96. The conclusion of this work was that metastable walks are a likelier explanation than straight-line long jumps along the bottom for the displacement distribution reported by Linderoth et al. [172]. Based on this study, it appears that so far mostly single jumps were observed in self-diffusion on the Pt(110)-(1 × 2) reconstructed surface.

3.3.4

Later studies At this stage, long jumps had been experimentally found only on channeled surfaces, on W(211). In 2002, however, Oh et al. [177] looked at the diffusion of palladium on W(110), to check if long jumps were limited to channeled planes and to provide a comparison with previous work. What had to be determined first was what sort of jumps might be expected on this surface. Possibilities, already depicted in Fig. 2.21, are single jumps to nearest-neighbor positions along the direction at the rate α, double jumps to next-nearest-neighbor positions in the same direction at the rate β, and horizontal jumps at the rate δx to the closest adsorption site along direction , as well as vertical transitions in direction at the rate δy .

130

Atomic events in surface diffusion

Fig. 3.94

Trajectories of Au adatom on reconstructed Au(110) surface with rows missing; obtained in molecular dynamics simulations using RGL interactions at 450 K. Left column: in-channel jumps single, double, and triple. Right column: metastable transitions single, double, and triple (after Montalenti and Ferrando [173]).

Fig. 3.95

Hard-sphere model of Pt(110)-(1 × 2) surface, showing metastable excursion of adatom on channel walls.

The Arrhenius plot for palladium on W(110) in Fig. 3.97 gives no indication of anything unusual – the experiments are nicely fitted by one straight line. At low temperatures, at T 185K, there are only small contributions from long jumps. By 210 K, however, for the distribution shown in Fig. 3.98, before any correction for

131

3.3 Long jumps in surface diffusion

Rate Γ(sec–1)

10–1

Single 0.92eV

10–2

Triple 0.98eV

10–3

Double 0.95eV

10–4 10–5 0.0026

Fig. 3.96

0.0028

0.0030 1/T (K–1)

0.0032

0.0034

Arrhenius plots for single length, double length, and triple length excursions of Pt adatom on Pt(110)-(1 × 2), obtained by density-functional calculations (after Lorensen et al. [176]).

Diffusivity Dx (10–16 cm2/sec)

200

Temperature T (K) 190 180

170

1 Pd on W(110)

10–1

10–2 ED = 0.509 ± 0.009 eV Do = 4.25(×1.7±1)×10–4cm2/sec

10–3 4.8

Fig. 3.97

5

5.2

5.4 5.6 1000/T (K–1)

5.8

6

6.2

Dependence of the diffusivity of single Pd adatom on W(110) upon the reciprocal temperature [177]. Data corrected for edge effects as well as for migration during transients.

transients in the temperature, the best fit is obtained with significant contributions of double as well as vertical jumps. The final evaluation, after correction for transients, gave β=α ¼ 0:12  0:06 and δy =α ¼ 0:11  0:10, where the errors are only rough estimates. These experiments for the first time demonstrated that various long jumps made significant contributions in two-dimensional diffusion even at relatively low temperatures, at less than 0.06Tm. What is worth noting is the huge difference between vertical and horizontal transitions for this system, which made it possible to derive a significant energy difference between the two kinds of jumps, as is illustrated in Fig. 3.99, that amounted to ~0.6 eV.

Atomic events in surface diffusion

Observations

132

210 K 0.7 sec = 3.35

1000 Observations

100

Best Fit Experiment

50

0 2 4

6 8

[110]

[001] Fig. 3.98

[111]

Displacement distribution of Pd adatom on W(110) plane at 210 K [177]. Double/single jump rate β=α is 0.12, δy =α is 0.11.

220

1

T (K) 200

190

180

ΔEδy ~ 0.6 eV

10–1 (–)/8t (sec–1)

210

Pd Experiments 10–2

10–3 (a) 10–4 10–1

Simulations

0 –10–1 (b) 4.4 Fig. 3.99

4.6

4.8 5 5.2 1000/T (K–1)

5.4

5.6

Temperature dependence of the difference in the displacement of Pd on W(110) along mean-square  x-and y-axis [177]. In simulations in (b) Δx2 ¼ Δy2 .

What was particularly surprising is that the vertical transitions are different from the horizontal transitions. How do they occur? Looking at Fig. 3.100, it is clear that atoms could jump directly along and directions, moving over a lattice atom, or else, a more likely alternative is that the atom starts along the energetically easiest

3.3 Long jumps in surface diffusion

Fig. 3.100

133

Schematic of jump rates on W(110) plane [179]. Basic jump is α transition along . At elevated temperatures, this can proceed beyond nearest-neighbor end point, giving β, δx , or δy.

direction, which is , and is then turned aside by interactions with the lattice. A turn in one direction produces a vertical jump. A turn in the other direction, to give a horizontal displacement, involves a larger deviation from the original direction of the jump and is therefore less likely. The same year Oh et al. [178] extended work to the self-diffusion of tungsten on the (110) plane of tungsten, and were able to demonstrate strong contributions from long jumps for this system. The trends were similar to what had previously been found for palladium diffusion. At 350 K, only vertical transitions made a contribution beyond nearest-neighbor single jumps, with a ratio δy =α  0:1. At 364 K the situation changed completely, as is apparent in Fig. 3.101. The ratio of rates β=α now amounted to 0.22, δx =α ¼ 0:36, and δy =α ¼ 0:43. Long jumps constitute roughly half the total number of transitions. It must be stressed here, however, that at elevated temperatures, adatom displacements take place during the rise in temperature before the temperature set for diffusion is reached, and also during the period at the end, after the heating current is shut off. The distribution of displacements occurring during these transients has to be separately determined in “zero-time” experiments, and the final rates are derived taking advantage of Eq. (2.12). The important question now remained, what are the energetics of long jumps? This question was addressed by Antczak [179] in 2004, who again worked with single tungsten atoms on the W(110) plane. The Arrhenius plot for the diffusion of tungsten is given in Fig. 3.102. Again there is nothing unusual in this plot, with an activation energy of 0.92±0.02 eV along the direction and almost the same barrier, 0.93±0.01 along the vertical . In the Arrhenius plot there is no clear indication of the long jumps which are known to be present. To get a true picture of what is happening, the distributions of displacements at several temperatures, as

Atomic events in surface diffusion

Observations

134

Best Fit Experiment

Best Fit β/α = 0.17 δx /α = 0.23 δy /α = 0.26

Observations

250 200

600 500 400 300 200

Best Fit Experiment

Best Fit 100 β/α = 0.04 0 δx /α = 0 δy /α = 0

150

[110]

100

[001] [111]

50 0

[110] [001] [111] Fig. 3.101

Distribution of tungsten displacements on W(110) at 364 K [179]. Best fit before corrections for “zero-time” effects obtained with significant contributions from long jumps. Inset gives distribution during temperature transients. Rates not corrected for effects from transients. Temperature T (K) 370

360

350

340

330

320

310

300

Dy ED = 0.93 ± 0.01 eV

Diffusivity D (10–16 cm2 sec–1)

10

Do = 5.21( × 1.4±1) × 10–3 cm2sec–1 1

W/W(110) 0.1 Dx ED = 0.92 ± 0.02 eV

0.01

Do = 1.55( × 1.6±1) × 10–3 cm2sec–1 0.001

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

1000/T (K–1) Fig. 3.102

Arrhenius plot for diffusivities of W atom on W(110) along and direction [179]. Best fit is obtained with straight lines.

well as the distributions during temperature transients consisting of at least 1000 observations each were measured. In order to understand what is going on here it is best to look at plots of the rates against 1/T. In Fig. 3.103 is shown the rate of single α jumps along directions. At low temperatures, the natural log of the single

135

3.3 Long jumps in surface diffusion

Temperature T (K) 1000

370

360 350

340

330

320

310

300

10

W/W(110) Eα = 0.94 ± 0.03 eV

νOα = 5.92

1

(× 2.5±1) × 1012 sec–1

10

0.1

1

0.01

0.001

0.1 Ir/W(110)

0.01

0.0001

Eα = 0.94 ± 0.02 eV 10–5

0.001

νOα = 5.45 0.0001

Rate of single W jumps α (sec–1)

Rate of single Ir jumps α (sec–1)

100

2.7

2.8

(× 1.9±1) × 1012 sec–1

2.9

3

3.1

3.2

3.3

3.4

1000/T (K–1) Fig. 3.103

Arrhenius plots for rates of single α jumps of W and Ir adatom on W(110) [11,179]. Single jump rate is fitted at low temperatures. At elevated temperatures, rate drops significantly below this straight line.

jump rate α follows nicely along a straight line. Above 340 K the measurements begin to fall below the Arrhenius curve, and at 365 K the rate α has actually turned down. This plot shows clearly that the number of single jumps is influenced by the presence of other jumps in the movement over the surface. This is not what happens with the rate of double jumps β, in Fig. 3.104, or the horizontal and vertical rates, in Fig. 3.105 and 3.106 respectively. They all show a linear dependence on reciprocal temperature over the investigated temperature range. We now have available in Table 3.3 estimates for energies and prefactors for all the tungsten jump rates. What is surprising is that the energy required to make long jumps is at least 30% higher than for singles, and the prefactors obtained for long jumps from the intersection of the Arrhenius plots are 103 higher than the standard factors. We can better understand what is going on here by abandoning the notion that the various jumps take place independently. At low temperatures, only nearest-neighbor transitions are possible, but that changes as the temperature is raised and correlations between different types of jumps become important. A possible explanation of the correlation is illustrated in Fig. 3.100. An adatom moving toward a nearest-neighbor site at a higher temperature (having a higher thermal energy) may continue to move, either to a nextnearest-neighbor position along giving a β jump, or the adatom path can deviate to the left to create a δy transition, or to the right, to form δx . In such a model the total

136

Atomic events in surface diffusion

370

Temperature T (K) 350 340 330

360

320

W/W(110) 100

1

10 1

0.01

0.1

0.001

0.01

0.0001

0.001 0.0001 10–5

Fig. 3.104

0.1

νOβ = 8.06 ( × 8.1±1) × 1015 sec–1

Ir/W(110) Eβ = 1.18 ± 0.04 eV

Rate of double W jumps β (sec–1)

Rate of double Ir jumps β (sec–1)

Eβ = 1.24 ± 0.13 eV

νOβ = 1.17 ( × 4.2±1) × 1015 sec–1 2.7

2.8

2.9 3 1000/T (K–1)

3.1

3.2

Arrhenius plots for the rate of double β jumps for W and Ir adatoms on W(110) [11,179].

Temperature T (K) 360

340

330

320

310

W/W(110) Eδx = 1.28 ± 0.13 eV

10 Rate of horizontal Ir jumps δx (sec–1)

350

1

0.1

νOδx = 3.73 ( × 8.6±1) × 1016 sec–1 1

0.01

0.1

0.001

0.01

0.0001

0.001

0.0001

10–5

Ir/W(110) Eδx = 1.25± 0.08 eV

Rate of horizontal W jumps δx (sec–1)

100

370

10–6

νOδx = 2.70 ( × 1.3±1) × 1016 sec–1 10–5 2.7

2.8

2.9

3

3.1

3.2

3.3

1000/T (K–1) Fig. 3.105

Temperature dependence of jump rates δx for Ir and W adatoms on W(110) plane [11,179].

137

3.3 Long jumps in surface diffusion

Table 3.3 Rate parameters for W adatoms on the W(110) plane [179]. Rate

Activation Energy ED (eV) Frequency Prefactor ν0 (sec−1)

α (low T) β δx δy

0.94±0.03 1.24±0.13 1.28±0.13 1.37±0.13

5.92(× 2.5±1)× 1012 8.06(× 8.1±1)× 1015 3.78(× 8.6±1)× 1016 1.00(× 4.4±1)× 1019

Temperature T (K) 360

340

330

320

310

W/W(110) Eδy = 1.37± 0.13 eV

10 Rate of vertical Ir jumps δy (sec–1)

350

1

0.1

νOδy = 1.00 ( × 8.4±1) × 1018 sec–1 1

0.01

0.1

0.001

0.01

0.0001

0.001

0.0001

10–5

Ir/W(110) Eδy = 1.27 ± 0.09 eV

Rate of vertical W jumps δy (sec–1)

100

370

10–6

νOδy = 4.61 ( ×

1.8±1)

2.8

2.9

×

1016 sec–1

–5

10

2.7

3

3.1

3.2

3.3

1000/T (K–1) Fig 3.106

Dependence of jump rates δy for W and Ir adatoms on W(110) plane upon reciprocal temperature [11,179].

number of jumps the adatom makes at any temperature is given by the number of basic jumps Rb in the direction. The different types of jumps originate from an attempt to make a transition to a nearest neighbor. The rate of long jumps R‘ can therefore be written as R‘ ¼ pb‘ Rb ;

(3:10)

where Rb is the rate of basic jumps, derived by extrapolation of the low temperature rate for single jumps to higher temperatures. The probability pb‘ of converting from a single jump can be represented as   X     E‘ Ej Eb‘ = ¼νb‘ exp
; (3:11) νj exp
pb‘ ¼ ν‘ exp
kT kT kT j

138

Atomic events in surface diffusion

where Eb‘ is the activation energy for this conversion, and j labels the various transitions possible. This makes the high activation energies and frequency factors for long jumps understandable. The activation energy E‘ for a given rate R‘ is now just the sum of the activation energies for the basic jump plus the transition, and the prefactor is the product of the individual prefactors. Furthermore, it is also clear why the single jump rate α diminishes at higher temperatures. Every time a long jump takes place, it diminishes the number of single jumps, since every long jump is a continuation of a single jump to a next-nearest-neighbor position, as illustrated in Fig. 3.100. Finally, if all the long jumps are related to nearest-neighbor transitions, in the way described above, then an Arrhenius plot of the sum of all the rates should plot as a straight line. As shown in Fig. 3.107, this is what actually happens, and the activation energy as well as prefactor are in reasonable agreement with the values found previously for singles alone at low temperatures as well as from the temperature dependence of the diffusivity. This all shows that at temperatures up to 370 K, the second-range transitions (double, vertical and horizontal jumps) replace single jumps in diffusion. It is likely that such a scenario will occur for longer transitions as well and that longer jumps will be eliminating shorter ones and taking over the leading role in transport at a certain temperature, gradually changing discrete motion into a continuous one along the channels on the surface. To check if this behavior is unique to self-diffusion, Antczak [11] has also looked at diffusion of iridium on W(110). As shown in Fig. 3.108, the activation energy for iridium

Temperature T (K) 100

370

350

330

310 10

W/W(110) Es = 0.89 ± 0.02 eV

1

νOs = 9.56 ( × 1.8 ) × 10 sec Sum of all Ir rates (sec–1)

±1

11

–1

1

0.1

0.01

0.1

0.001

0.01 Ir/W(110) Es = 0.90 ± 0.02 eV

0.001

Sum of all W rates (sec–1)

10

0.0001

νOs = 1.69 ( × 2.0±1) × 1012 sec–1 0.0001

Fig. 3.107

2.7

2.8

2.9

3 3.1 1000/T (K–1)

3.2

3.3

10–5 3.4

Arrhenius plot of the sum of all jump rates for W and also Ir adatoms on W(110), giving a straight line fit [11,179].

139

3.3 Long jumps in surface diffusion

370

350

Temperature T (K) 330

310

Dy

Diffusivity D (10–16 cm2 sec–1 )

10

ED = 0.94 ± 0.02 eV Do = 6.78 ( × 1.8±1) × 10–3 cm2 sec–1

1

0.1 Dx 0.01

ED = 0.94 ± 0.02 eV Do = 3.00 ( × 1.8±1) × 10–3 cm2 sec–1

0.001 2.7

Fig. 3.108

2.8

2.9

3 3.1 1000/T (K–1)

3.2

3.3

3.4

Arrhenius plot for Ir adatom diffusivities on W(110) plane along and [11].

diffusion is very similar to tungsten.4 The activation energy for motion along and is now the same, 0.94±0.02 eV, and the prefactor along is roughly a factor of two larger than p along ffiffiffi , as it should be since the inter-atomic spacing in the direction is 2 larger. The distribution measured at a low temperature, at 301 K, shown in Fig. 3.109, is essentially due to single jumps between nearest-neighbor positions. As the temperature is raised, the contribution from long jumps increases, as already seen previously. In Fig. 2.23 is shown the displacement distribution at 360 K, together with the distribution obtained for “zero-time” measurements to catch transitions during temperature transients. The contribution from long jumps has now become significant, with a best fit to the experiments obtained for β=α ¼ 0:15; δx =α ¼ 0:38, and δy =α ¼ 0:32. This trend continues as the temperature is raised further. All the jump parameters for Ir are given in Table 3.4. If the jump rates for iridium are compared with those previously established for tungsten on the same plane they are found to be quite similar, as can be seen in Figs. 3.103–3.107. Activation energies as well as prefactors for long jumps of iridium are again high. The single jump rate α, shown in Fig. 3.103, again decreases as the diffusion temperature reaches higher values, and just as previously, the sum of all the jump rates plots linearly on an Arrhenius plot in Fig. 3.107, with an activation energy very close to that of single jumps obtained at low temperatures and also derived from the 4

The possibility of exchange was ruled out by measurements of the evaporation voltage after diffusion.

140

Atomic events in surface diffusion

Table 3.4 Rate parameters for Ir adatoms on the W(110) plane [11]. Rate

Activation Energy ED (eV)

Frequency Prefactor ν0 ðsec
1 Þ

α(low T) β δx δy

0.94±0.02 1.18±0.04 1.25±0.08 1.27±0.09

5.46(× 1.9±1)× 1012 1.17(× 4.2±1)× 1015 2.70(× 1.3±1)× 1016 4.61(× 1.8±1)× 1016

800 700

Best Fit α = 0.00094 β/α = 0 δ x /α = 0 δ y /α = 0

Observations

600

Experiment

500 400

Best Fit

300 200 100 0

[110] [001] [111]

Fig. 3.109

Distribution of Ir atom displacements on W(110) at 301 K, revealing only nearest-neighbor jumps [11].

diffusivity. The one big difference is that the vertical δy and the horizontal δx jump rates are now essentially equal. It turns out that the difference in the activation energy required for vertical and horizontal jumps on W(110) depends on the chemical identity of the adatom moving over the surface. The energy difference as a function of the atomic mass of the adatom is plotted in Fig. 3.110 and gives a linear dependence [180]. Lighter atoms like Pd very strongly feel the anisotropy of the W(110) surface, while heavier atoms like Ir are not

141

3.3 Long jumps in surface diffusion

0.7 Pd 0.6

Eδy – Eδx (eV)

0.5 0.4 0.3 0.2 W 0.1 Ir 0 –0.1 100

Fig. 3.110

120

140 160 Atomic Mass

180

200

Difference in activation energies for transitions along y- and x-axes of W(110) as a function of adatom mass [180].

much influenced. So far information is available for only three types of adatoms; further investigation is definitely necessary. On W(110), in addition to single jumps, a second range of transitions including β double, vertical δy , and horizontal δx was detected and it was demonstrated that this second-range transition started to play a leading role in the movement of atoms over the surface at quite a low temperature, around 1/10 of the melting point. It is just possible that at higher temperatures, longer transitions will in turn start to be present in the movement of adatom. Both iridium and tungsten atoms show much the same behavior in diffusion on W(110). Can this similarity be transcribed to other surfaces as well? To check this, self-diffusion has also been examined by Antczak [181] on the one-dimensional W(211) plane, on which a tungsten atom diffuses in a single channel. Displacement distributions were measured at increasing temperatures, and were later corrected in “zero-time” experiments. At 295 K only atom jumps between nearest-neighbor sites were detected. However, at 300 K and higher temperatures, jumps spanning two spacings along became more important, as is clear from Fig. 2.22. Up to a temperature of 325 K no triple jumps were detected, but they could be active at higher temperatures. The rates of single jumps, adjusted for “zero-time” results, are shown in Fig. 3.111; what is striking here is that for temperatures above 300 K, the rate falls very rapidly below the values extrapolated from lower temperatures. For the lower temperatures, the activation energy for single jumps is 0.84±0.06 eV, with a frequency prefactor of 2.2 (× 11.3±1)× 1013 sec−1, close to what was derived from the temperature dependence of

142

Atomic events in surface diffusion

Temperature T (K) 320

300

260

Eα = 0.84 ± 0.06 eV

1 Rate of single jumps α (sec–1)

280

ν0α = 2.2 ( × 11.28±1) × 1013 sec–1

0.1

0.01

W/W(211) 0.001 0 3

Fig. 3.111

3.2

3.4 3.6 1000/T (K–1)

3.8

Arrhenius plot for α single jumps of W on W(211) [181]. At T > 320 K, rate α has decreased so much it is no longer discernable.

the diffusivity, shown in Fig. 2.20. The rate of double jumps, plotted in Fig. 3.112 for the different temperatures, gives a reasonable straight line, with an activation energy of 1.44 ±0.13 eV and a much larger prefactor, 7.9(× 127±1)× 1021 sec−1, effects similar to what was found on W(110). As was already discovered in the self-diffusion of tungsten atoms on W(110) above 350 K, the single jump rate α descends below that extrapolated from low temperatures, while the other rates β, δx , and δy do not – they follow normal Arrhenius plots. The explanation for diffusion on W(211) is similar to that for W atoms on the W(110) plane – the transitions are not independent and all arise from one basic jump process. At higher temperatures single jumps are gradually replaced by longer transitions. The sum of all the rates on W(110) with both tungsten and iridium adatoms yielded good Arrhenius plots, with characteristics similar to what was derived for diffusivities. When the sum of single and double jumps detected on W(211) is plotted against 1/T, as in Fig. 3.113, the situation is seen to be different: at higher temperatures the sum falls significantly below the line representing the total number of jumps extrapolated from low temperatures. Double jumps evidently do not replace single jumps completely. As indicated in the model in Fig. 3.114, an atom jumping at a nearest-neighbor site can do three things. It can equilibrate at that site, giving a single transition. It may also continue to a site two spacings away, in a double jump, or else it can rebound and return to the origin, to the starting position. Such rebounds have been detected in molecular dynamics

143

3.3 Long jumps in surface diffusion

Temperature T (K) 325

320

315

Rate of double jumps β (sec–1)

1

310

305

300

Eβ = 1.44 ± 0.13 eV

ν0β= 7.90 ( × 127.31±1) × 1021 sec–1

0.1

0.01 W/W(211)

0.001 3.05

3.1

3.15

3.2

3.25

3.3

3.35

–1

1000/T (K ) Fig. 3.112

Normal Arrhenius plot for β double jumps of W adatom on W(211) [181]. Temperature T (K) 300 280

320

260

Eα = 0.84 ± 0.06 eV

Sum of jumps α + β (sec–1)

1

0.1

W/W(211)

0.01

0.001 3

Fig. 3.113

ν0α = 2.2 ( × 11.28±1)×1013 sec–1

3.2

3.4 1000/T (K–1)

3.6

3.8

Arrhenius plot for the sum of all jump rates measured for W adatom on W(211); for temperatures  310 K, rate falls below linear plot [181].

144

Atomic events in surface diffusion

β

α βR Fig. 3.114

Schematic of different types of jump processes for an adatom on W(211) [181]. βR denotes rebound transition.

Temperature T (K) 310

320

300

Rate of rebound jumps βR (sec–1)

EβR = 1.03 ± 0.06 eV

νβR = 1.4 ( × 10.3±1) × 1016 sec–1

1

0.1

0.01 3.05

W/W(211)

3.1

3.15

3.2

3.25

3.3

3.35

3.4

1000/T (K–1) Fig. 3.115

Rate of rebound jumps, obtained as the difference between the sum of single plus double jumps and total jump rate extrapolated from low temperatures [183].

simulations of diffusion on a bcc(211) as well as on fcc(111) planes [158,165,182] and do not result in a displacement but they cost energy. That is why it is important to include them in an atomistic description of diffusion. These rebound transitions would not be seen directly in the distribution of displacements as their displacements are zero. Nevertheless, Antczak [183] has for the first time ever also been able to determine the characteristics of the rebound transition. They can be found as the difference between the total rate of jumps, obtained by extrapolating the jump rate from low temperatures, and the sum of single plus double transitions. This rate of rebounds is plotted in Fig. 3.115 as a function of the reciprocal temperature; rebounds

145

3.3 Long jumps in surface diffusion

Table 3.5 Rate parameters for W adatoms on the W(211) plane [183].

Fig. 3.116

Rate

Activation Energy ED (eV)

Frequency Prefactor ν0 ðsec
1 Þ

α(low T) β βR

0.84 ± 0.06 1.44 ± 0.13 1.03 ± 0.06

2.2(× 11.3±1)× 1013 7.9(× 127±1)× 1021 1.40(× 10.3±1)× 1016

Model showing adatom movement on W(211) as lattice atoms at the sides move out of the way [183].

occur with an activation energy of 1.03±0.06 eV and a frequency prefactor of 1.40 (× 10.3±1)× 1016 sec−1. That is, the characteristics for rebounds lie between those of single and double jumps on W(211). The existence of rebound jumps is proof of how important the dynamic of the lattice are: rebounds are caused by the correlation between adatom movement and the vibrations of the lattice. That is also clear from Fig. 3.116, showing how an atom moves down a channel as lattice atoms at the sides open up. The energetics of all jumps in self-diffusion on W(211) up to 325 K are presented in Table 3.5, but the nature of the jumps will probably change as the temperature is increased. Here we just want to note briefly that long jumps have also been observed for entities larger than adatoms. The first experimental indications of long jumps of bigger clusters with an open structure – Re5 diffusing on the W(110) surface – was obtained by Fink in 1985 [184]. They also were seen for Ir19 clusters on Ir(111) surface by Wang et al. [185] and later for decacyclene and hexa-tert-butyl decacyclene by Schunack et al. [186]. Indications of long jumps were similarly found in molecular dynamics simulations of diffusion along straight steps of Cu(111) by Marinica et al. [187], shown in Fig. 3.117, but this has not yet been experimentally investigated. What is clear from this survey is that long jumps play an important part in surface diffusion, especially at elevated temperatures. Observations of long jumps turn out not to be limited to certain surface structures. The temperature for detecting these transitions is relatively low, so that the existence of long jumps has to be taken into account as one of the significant events on the surface. Nearest-neighbor jumps, which are widely used in modeling surface processes, disappear almost completely at relatively low temperatures. That at least is the case for diffusion on tungsten. What must still be established now is that similar effects occur on other materials, but this is a task for the future.

146

Atomic events in surface diffusion

single jump

double jump

i +1 i i –1 i –2

1500

rejected jump 1550

1600 1650 time (psec)

1700

1750

Fig. 3.117

Time evolution of the position of a Cu adatom at a step on Cu(111) plane, showing single, double, and rejected jumps. Results from simulations based on EAM potentials (after Marinica et al. [187]).

3.4

Transient diffusion So far we have concentrated on the atomic processes significant in diffusion in a surface layer in thermal equilibrium. This, however, is not the only way in which diffusion may occur. When atoms strike a surface from the vapor and are trapped there, the energy of condensation still has to be transferred to the lattice. Part of this energy may activate the trapped atom to diffuse over the surface, in what traditionally is referred to as transient diffusion. This is a subject that has long been of interest, as transient diffusion could be important in reactions at low temperatures, where thermal diffusion is stopped.

3.4.1

Transient motion of adatoms Experiments to detect the diffusion of atoms after striking a cold surface are not easy and require modern surface analytical techniques, but there were rough, early studies made to explore this phenomenon. The first attempts [188–191] to more seriously come to grips with these problems, around 1960, were usually theoretical in nature. Estimates were made for atom collisions with the end of a linear chain of atoms at rest, that is, with a onedimensional crystal. A discussion of the behavior of atoms after trapping was given by McCarroll [192], who, as shown in Fig. 3.118, determined the time such an atom took to come to equilibrium. It was found that after only three vibrations, 98% of the energy of condensation had been transferred to the one-dimensional lattice. Of course these onedimensional models provide only a crude pattern for the properties of real systems, but experiments began to get started. In 1963, a tungsten emitter tip in a field ion microscope was shadowed from the side with tungsten atoms, as depicted in Fig. 3.119, and was then imaged [190,191,193]. For extensive transient diffusion, a fairly uniform layer over the surface would be expected. Instead what was found is shown in Fig. 3.120. Some atoms penetrated into the (110) region, but only a few. In view of the poorly defined geometry of the evaporator in relation to the emitter, the only thing clear is that transient diffusion of tungsten on a tungsten surface has at best quite a limited range.

147

Displacement From Equilibrium Bond Length

3.4 Transient diffusion

ξ=1 κ=1

ξ = 0.4 κ=1

1

ξ=1 κ = 0.5

0 –1

Energy Equilibration After Trapping

–2 K

–3

3

2

M

K0

M0

1

0

(0)

r

0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30 32 ϖot

Fig. 3.118

Time for equilibration of energy in collision of different gas atoms with one-dimensional lattice [191,192]. ξ is the ratio of the mass for an incident compared with a lattice atom,  is ratio of force constants.

Fig. 3.119

Schematic of deposition arrangement for metal atoms in FIM [193].

Further experiments on tungsten were undertaken by Gurney et al. [194] in 1965. They again shadowed with tungsten atoms one side of a tungsten emitter tip, kept at a low temperature, achieving a coverage of one to two layers. Field ion micrographs and field evaporation showed small piles of atoms, as in Fig. 3.121. The nature of the deposits was analyzed in detail by Young and Schubert [195], using Monte Carlo simulations for the deposits shown in Fig. 3.122. Their models assumed two different scenarios: atoms condensing at the first site struck, shown in Fig. 3.122a, and condensation following two random jumps, as in Fig. 3.122b. Statistical estimates for both showed a lowering of the density of atoms in upper layers for transient hopping, contrary to the experimental observations. They concluded, that tungsten atoms condensing on a tungsten surface at a temperature below 77 K are bound in the first potential well found.

148

Atomic events in surface diffusion

Fig. 3.120

FIM images of sideways deposition of W atoms on W(110)-oriented tip at ~20 K [190]. (a) Clean surface, with direction of atoms shown by arrow. (b) After atoms have been deposited from source at ~3000 K.

Fig. 3.121

FIM image of tungsten deposit after some field evaporation. White frame indicates area analyzed in Monte Carlo simulations (after Gurney et al. [194]).

This finding was bolstered in 1980 when Flahive and Graham [6] deposited tungsten atoms onto a W(111) surface in a field ion microscope. In 220 adsorption observations on a plane of 24 sites the atom was deposited 156 times into a lattice site and 64 times close to the edges. Adatoms were not found at fault sites. Flahive and Graham analyzed their findings with respect to edge atoms and compared them with the distribution obtained for random deposition of adatoms. As it is clear from the results in Table 3.6, a random distribution very well matched experimental observations. Such a random distribution is expected if the colliding atoms settled at the first site encountered. Some time later, in 1985, Tully [196] turned his attention to the interaction of xenon atoms with a Pt(111) surface. Xenon, of course, interacts only weakly with the surface, primarily through van der Waals forces, and can be expected to behave differently from strongly binding tungsten atoms. Tully did stochastic trajectory calculations which

149

3.4 Transient diffusion

Table 3.6 Distribution of W atoms deposited on W(111) surface [6].

Fig. 3.122

Adatom surrounding surface atoms which are (111) edge atoms

0

1

2

Observations Frequency Random distribution

48 0.31 0.42

38 0.24 0.23

70 0.45 0.35

Results of Monte Carlo simulation of tungsten deposition. (a) Condensation on impact. (b) Condensation after two random jumps following impact. Black circles indicate position of first layer obtained in simulations (after Young and Schubert [195]).

established that around 40% of the impinging xenon atoms were trapped briefly on the surface at a temperature of 773 K. The population of trapped atoms decayed exponentially due to thermal desorption, as shown in Fig 3.123a, the z-component of the energy being very quickly quenched. But exchange of energy between atom and lattice is clearly seen in Fig. 3.123b. The trapped atoms retain some of their velocity component parallel to the surface, and drift ~ 200 Å along the surface. In this instance, energy exchange between the incoming atom and the surface was obviously limited and there seemed to be considerable transient diffusion. In the same year, Schneider et al. [197] carried out molecular dynamics simulations with a Lennard-Jones system to describe epitaxial growth at different temperatures of a (111) substrate. Even at temperatures approaching zero they found the deposited layer to consist of nicely ordered close-packed islands, as in Fig. 3.124. The conclusion drawn from these results was that atom diffusion took place on deposition, even at very low temperatures. However, Schneider et al. [197] had just one movable layer in their simulation, and it later turned out that this was responsible for their findings. Sanders et al. [198] discovered that for such a model the adatom was subject to multiple kicks by the substrate, which does not describe the nature of the deposition correctly. Attention returned to experiments the next year with the work of Fink [199,200] on the condensation of tungsten atoms. In a field ion microscope, he attempted to deposit a

150

Atomic events in surface diffusion

1.0

F(t)

(a) 0.5

0 (b)

<ET >/K

2000

1000 773 K

/104 cm sec–1

0 (c)

3 2 1 0

0

20

40

60

80

t /10–12 sec Fig. 3.123

Thermalization of Xe incident at 45° on Pt(111) at 773 K. (a) Fraction of atoms on the surface as a function of time. (b) Average kinetic energy in x-direction (dashed), in y-direction (dot-dashed) and along z, the surface normal, for atoms on the surface. (c) Average x-component of velocity of adsorbed Xe (after Tully [196]).

single atom on a (111) plane of tungsten that had been reduced to just three atoms. It turned out that deposition took place readily, as illustrated in Fig. 3.125. If a tungsten atom trapped on the surface after deposition were to undergo lateral translations, such deposition would be highly unlikely; atoms striking the small (111) plane would move over the edges without any chance of returning. Fink’s conclusion therefore was that the condensing tungsten atom settled into the first site struck. It must be noted, however, that a barrier at the step edge could have helped to confine the atom. In 1989 there appeared an important paper by Egelhoff and Jacob [201] which created a great deal of interest. They looked at reflection high energy electron diffraction (RHEED) oscillations during growth at a low temperature of 77 K for copper and iron on Cu(100), as well as Ag, Cu, Fe, and Mn on Ag(100).5 If condensation occurs at the first location struck, no layer growth was expected. Egelhoff and Jacob found oscillations, as shown in Fig. 3.126, and concluded that quasi layer-by-layer growth was achieved even at 77 K, where thermally activated diffusion was not possible. “The deposited atoms,” they claimed, “must be using their latent heat of condensation 5

Oscillations were found even earlier by Koziol et al. [202], who examined the deposition of nickel on the W(110) plane at 100 K.

3.4 Transient diffusion

151

Fig. 3.124

Location of atoms in first layer of Lennard-Jones system after deposition of particles on surface at T = 0 K of movable substrate (after Schneider et al. [197]).

Fig. 3.125

FIM images of W atom deposition onto a W trimer on W(111) plane. (a) Trimer formed by field evaporation. (b) W atom has been deposited on trimer, and now is the only atom seen in center. (c) Superposition of (b) on image after removal of W adatom. Additional deposited atoms are now visible (after Fink [199]).

( 3 eV) in lieu of thermally activated diffusion to overcome the energy barrier to surface diffusion.” In light of what had previously been found for the immediate localization of tungsten atoms during condensation, this result was surprising indeed. It must be noted, however, that LEED intensity oscillations were also found by Flynn-Sanders et al. [203] for palladium on Pd(100) even at 100 K, as shown in Fig. 3.127. Evans et al. [204] in 1990 offered an ingenious proposal to account for layer-by-layer growth previously observed at low temperatures by Egelhoff and Jacob [201]. Rather than invoking transient diffusion to account for the necessary mobility, their model involved “downward funneling to complete adsorption sites of all impinging atoms at T = 0 K,” as illustrated in Fig. 3.128. “Such downward funneling,” they reported, “might correspond to channeling down the (111) faces of micropyramids which developed during film growth.” With this model they obtained Bragg peaks oscillating when the number of layers increased, as found in the various experiments mentioned. At this stage the overall conclusion was that transient diffusion played at best a very small role in condensation of metal atoms on a metal surface.

152

Atomic events in surface diffusion

100

60 sec

Relative Intensity (%)

(c) Cu/Ag(100) T = 77 K

50

100 120 sec

(b) Fe/Ag(100) T = 300 K

50 100

120 sec

(b) Fe/Ag(100) T = 77 K

50 0 Fig. 3.126

1

2

3 4 5 6 7 ML Deposited

8

Oscillations in RHEED intensity from layers of various metals deposited on low temperature substrates (after Egelhoff and Jacob [201]).

This impression was strengthened in 1991 by the work of Wang [20], who looked at the condensation of iridium, rhenium, tungsten, and palladium atoms on the Ir(111) plane, using the field ion microscope to identify the location of adsorbed atoms. The lowest surface temperature in these experiments was ~ 20 K, but deposition was also studied for increasingly warmer surfaces. As has already been pointed out, on the fcc(111) plane, two types of binding sites exist, fcc and hcp, at which atoms have slightly different binding energies. For iridium atoms, binding is 0.022 eV stronger at hcp than fcc sites, for rhenium the difference is 0.14 eV, and for tungsten it amounts to 0.19 eV [22]. The distribution of the different atoms over the sites of the (111) plane found in the experiments is shown in Table 3.7. The atom, at a temperature TE, striking the surface at a temperature TS, is found at sites ordinarily not significantly populated during thermal motion. It is clear that at a low temperature of ~ 20 K, the distribution over the surface is random and without any preference for the more strongly binding sites, even if such preference is observed during diffusion. The conclusion was “that rapid localization at the point of first impact is fairly common for strongly bound metal atoms, even on as smooth a surface as Ir(111).” Layer-by-layer growth at low temperature stimulated more refined calculations of the dynamics of atom condensation on metals. Sanders and DePristo [205] simulated condensation on fcc(100) layers at 80 K, relying on corrected effective medium potentials.

153

3.4 Transient diffusion

(0,–1)

(1,–1)

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0 1

0 1

100 K

200 K

300 K

400 K

500 K 0 0 0 250 0 250 Cumulative Deposition Time (sec) Fig. 3.127

Oscillations of Bragg intensity from Pd layer on Pd(100) as a function of the amount deposited. Shown are results for two diffraction spots under out-of-phase conditions (after Flynn-Sanders et al. [203]).

Fig. 3.128

Schematic diagram showing (a) random deposition and (b) downward funneling deposition on a surface (after Evans et al. [204]).

One of the first things they checked was the importance of the number of movable atom layers in the model of the substrate crystal. They discovered that on increasing the number of substrate layers with movable atoms to three, the trapping of incoming atoms happened usually at the first site encountered on their fcc(100) substrate. For nickel on Ni(100), rhodium on Rh(100), palladium on Pd(100), platinum on Pt(100), and gold on Au(100) they reported the absence of transient diffusion, as is suggested by the examples in Fig. 3.129. Only for silver on Ag(100), with quite a small calculated diffusion barrier, were there any signs of atoms beyond the unit cell at which the

154

Atomic events in surface diffusion

Table 3.7 Distribution of deposited metal atoms over sites on Ir(111) [20]. At fcc sites Atom

TE (K)

TS (K)

Number deposited

Random

Actual

%

Ir

2500

20 80 100 104 20 120 140 200 215 20 100 200 20 65

346 104 102 503 507 319 158 106 265 248 213 285 216 175

173±9 52±5 51±5 252±11 254±11 160±9 79±6 53±5 133±8 124±8 107±7 143±8 108±7 88±7

178 54 20 71 253 158 26 0 0 120 103 0 116 175

51.4 51.9 19.6 14.1 49.9 49.5 16.5 0 0 48.4 48.4

Eαf ~ 0.24 a Eαh ~ 0.28 Re Eαf ~ 0.39 Eαh ~ 0.52

3000

W Eαf ~ 0.30 Eαh ~ 0.48 Pd Eαf ~ 0.17 Eαh ~ 0.13

3200

a

1700

53.7 100

Unit = eV; Eαf – activation energy for jump out of fcc site; Eαh – activation energy for jump out of hcp site

Ni/Ni(100)

Rh/Rh100)

Pt/Pt100)

2

2

2

1

1

1

0

0

0

–1

–1

–1

–2 –2

–1

0 1 Pd/Pd(100)

2

–2 –2

–1

0 1 Ag/Ag(100)

2

–2 –2

2

2

2

1

1

1

0

0

0

–1

–1

–1

–2 –2 Fig. 3.129

–1

0

1

2

–2 –2

–1

0

1

2

–2 –2

–1

–1

0 1 Au/Au(100)

0

1

2

2

Disposition of atoms condensed on (100) planes of different metals at 80 K. Calculations done with MD/MC-CEM potentials show no transient diffusion (after Sanders and DePristo [205]).

3.4 Transient diffusion

Fig. 3.130

155

Distribution of Cu atoms deposited on Cu(111) plane at 80 K. Only 67% of gas atoms initially concentrated over central cell end in central cell on the surface. Simulations done with MD/MCCEM potentials. Results suggest some transient motion (after Sanders et al. [198]).

incoming atoms had been aimed. Sanders et al. [198] somewhat later did similar calculations for condensation on the energetically much smoother fcc(111) plane, again at 80 K. They report “no transient mobility for the Cu/Cu(111) system,” but their results, obtained with the corrected effective medium theory, revealed that 67% of the condensing atoms settled in the first unit cell struck. Of the rest, 21% were found in the adjacent unit cells, 11% two units away, and the remainder three. Their distribution is shown in Fig. 3.130. In 1992, Weiss and Eigler [206] reported a very important experiment they had carried out on the adsorption of xenon on Pt(111). Their work was done in a low temperature scanning tunneling microscope with the (111) surface at 4 K and illuminated with a sharply collimated beam of xenon atoms at a temperature between 100 and 300 K. At a low coverage of 0.01 monolayers, the xenon atoms appeared at steps at the edges of the plane, as in Fig. 3.131a. The authors concluded that the atoms move over hundreds of angstrom units over the surface to reach the steps and lose their excess energy. On increasing the Xe coverage to 0.03 ML, Xe islands such as in Fig. 3.131b began to appear. After moving an island away a depression could be observed in the Pt(111) surface associated with the defect on which nucleation started. This finding definitively demonstrated transient diffusion for weakly binding xenon atoms, just as predicted seven years earlier by Tully [196]. It is of interest to note that this long diffusion path is tied to the smoothness of the Pt(111) plane. No such effect was reported for xenon on Ni(110) [207] kept at 4 K, made up of close-packed rows of atoms. A number of epitaxial growth experiments have also been carried out, and Ernst et al. [208] found oscillations in the scattering intensity with the amount of copper deposited on Cu(100) down to 100 K. Much the same was reported by Bedrossian et al. [209] for silver deposits on Ag(100), where oscillations were again found at 100 K. In both papers transient diffusion as well as downward funneling was mentioned as a possible

156

Atomic events in surface diffusion

Fig. 3.131

STM image of Pt(111) bombarded by xenon atoms. (a) Step decorated with Xe atoms that had been deposited on the surface. Vertical scale is expanded. (b) Image of Xe island on Pt(111) obtained with STM (after Weiss and Eigler [206]).

explanation for the apparent crystal growth. No final decision was reached about which one is really responsible for these oscillations. Blandin and Massobrio [210] in 1992 looked at the collision of silver dimers, Ag2, with the Pt(111) surface in simulations relying on EAM interactions. Their results for the displacement of Ag atoms from the point of impact for dimers colliding with an energy of 0.5 eVand also 5 eVare shown in Fig. 3.132. Even for 5 eV impacts, the distance from the point of impact was at most 5 Å, indicating the absence of transient motion. It is interesting to note that on colliding with the surface the dimer had a considerable likelihood of dissociating. With the surface at 150 K, the fraction of Ag2 dissociating for 0.5 eV impacts was 0.29.6 Condensation on a Lennard-Jones fcc(111) plane, with the lattice atoms initially at rest, was looked at in greater detail by DeLorenzi [211], who found condensed atoms out through the third shell of sites surrounding the aiming point. This result is illustrated in Fig. 3.133. The details of the energy transfer process in condensation were also examined. Atoms that impinged directly on atomic binding sites generally condensed right there, but collisions on top of a surface atom sent the incoming atom far out, as suggested in Fig. 3.134. The process of energy transfer, illustrated in Fig. 3.135, is different in the two cases. After collision with a binding site the energy wave created on impact is quite gentle; the energy of the impinging atom is mostly transferred between three surface atoms. The rebound energy Er the adatom gets back from the surface is small, not enough to cause further movement. The situation changes in the collision of an adatom with a surface atom. The head-on-collision influences mostly one surface atom and the atom immediately underneath it. The rebound energy Er the adatom receives in the lattice response to the impact is bigger, big enough to cause the movement of the atom over a few atomic distances, as shown in Fig. 3.133. The conclusion from all these simulation studies was that on the energetically smooth fcc(111) plane the only transient diffusivity 6

At 300 K, dimers dissociate on impact but recombine during further movement on the surface.

157

3.4 Transient diffusion

0.6

dd (10–7 cm)

0.5 0.4 0.3 0.2 0.1 0

0

2

4 t

(10–12

6

8

10

sec)

Fig. 3.132

Average distance between point of impact of Ag2 with Pt(111) and resting place on the surface, established in EAM calculations. Impact kinetic energy: open circles 5 eV, full circles 0.5 eV (after Blandin and Massobrio [210]).

Fig. 3.133

Location of atoms before and after condensing on fcc (111) surface at T = 0 K [211]. Results obtained from molecular dynamics simulations with Lennard-Jones potentials.

observed can result from head-on collisions; even there, atoms are confined close to the site of the initial impact, but it is necessary to remember that this investigation was on a Lennard-Jones crystal, not for a specific material. Experiments on the condensation of iridium atoms on the surroundings of small Ir clusters formed on the Ir(111) plane were carried out by Wang and Ehrlich [212] in 1993. Even when the surface temperature was kept at ~ 20 K, a zone empty of any deposited atoms was observed, such as that around the Ir59 cluster shown in Fig. 3.136. Rather than invoking transient diffusion as an explanation, this empty zone was attributed to changes in the potential binding adatoms close to a cluster

158

Atomic events in surface diffusion

–6 –4 –2 3

x/xO 0 2

4

6

8

–6 –4 –2 3 (b) Z/ZO

1

4

4

y/yO

8

4

6

8

initial position

2

condensation site

0

–2

6

1 0

2

4

2

0

y/yO

Z/ZO

(a) 2

x/xO 0 2

0

–2

initial position

condensation site –4

–4 –6

Fig. 3.134

–4 –2

0 2 x/xO

4

6

8

–6

–4 –2

0 2 x/xO

Trajectories for atoms impinging on fcc(111) plane in two situations [211]. (a) Collision at hcp site. (b) Collision on top of surface atom.

edge, as shown in Fig. 3.137. This was described as unphysical by Kellogg [213]. However, such distortions have been reported in calculations by Liu and Adams [214], Stumpf and Scheffler [85], Villarba and Jónsson [71], as well as by Jacobsen et al. [215], eliminating the possibility that transient diffusion occurs for atoms colliding with the cold surface close to the cluster. The problem of the empty zone will be described more extensively in Chapter 6. It should be noted, however, that in STM experiments on nickel films deposited on Au(110)-(1 × 2) at 130 K, no indication of empty zones was found by Hitzke et al. [216]. Recently Smirnov et al. [39] relied on ab initio calculations and kinetic MC simulations to suggest that the presence of an empty zone around a cluster is associated with quantum confinement of surface electrons. Vandoni et al. [217] in 1994 carried out studies of helium atom scattering while a monolayer of 0% to 10% silver was deposited on Pd(100). The surface was maintained at 80 K, a temperature at which Ag adatoms were not mobile. As shown in Fig. 3.138, the experimental results for the measured intensity could not be fitted over the entire range of concentrations deposited on the assumption of complete accommodation of the Ag at the first site struck. Agreement could be obtained assuming six random consecutive jumps prior to permanent adsorption at a site. It should be noted,

159

3.4 Transient diffusion

(a)

(b)

αa1 αa2

2

βa

0

2 Adatom 0 Er

Adatom

γa

–2

Plane

0 0

αa2

1

0

αa1 βa

2

αa1 αa1

0 γa

3

1

ΔE(ε)

ΔE(ε)

0

βa γa

Er

–2 Plane

0

αa1 αa2

0

Σ

2

βa

3

γa

Σ

0 Eads

0

100

200

Time (MD-steps) Fig. 3.135

300

Eads

0

100

200

300

Time (MD-steps)

Energy changes of colliding and lattice atoms during collision [211]. (a) Collision at hcp site. (b) Collision with a lattice atom.

however, that up to a coverage of 2% good agreement was obtained for instantaneous condensation. As an alternative model, Vandoni et al. assumed that an atom landing close to an already deposited atom or cluster would join up. To obtain good agreement with the measured intensity it had to be assumed, as shown in Fig. 3.139, that an atom landing one site remote would combine with its neighbor despite the low temperature. The authors conclude, however, that “the lowering of the diffusion barrier due to the presence of an adsorbate combined with a transient mobility seems to be the most probable explanation for these experimental findings.” A year later the same authors [218] tackled the same problem once more. This time, in addition to MC simulations they used EAM potentials to carry out molecular dynamics simulations. They performed 50 adatom depositions with an energy from 0.16 to 2.98 eV. Some 6% of the adatoms moved towards already adsorbed adatoms, and no transient mobility of atoms was observed. However, these results did not explain their experimental data. They claimed that it was probably due to an inaccurate potential or an overestimation of energy dissipation in EAM. From molecular dynamics simulations they concluded that a 6.1 Å capture zone existed around preadsorbed adatoms and that lowering of the potential barrier combined with transient mobility was the most probable scenario.

160

Atomic events in surface diffusion

(a)

(b)

24 16 8 0 –8 –16 (c) –24 –48 –36 –24 –12

0

12

24

36

48

Fig. 3.136

Ir atoms deposited on Ir(111) plane at ~20 K with Ir59 cluster at the center [212]. (a) FIM image with island in place on Ir(111) plane. Arrow shows direction of incoming atom. (b) Single Ir atom has been deposited on surface. (c) Distribution of deposited atoms around central island, showing empty ring.

Fig. 3.137

Potential for adatom migrating over Ir(111) plane with a central cluster on it [212]. Bottom curve is adequate for diffusion results, top gives more severe changes required for results in condensation experiments.

161

3.4 Transient diffusion

1.0 measured 6 steps 1 step Ig Intensity l/l0

0.8

0.6

0.4

0.00

0.05

0.10

Coverage (ML) Fig. 3.138

Intensity of reflected helium for Ag deposits on Pd(100). Best fit obtained with six random transient steps (after Vandoni et al. [217]).

0.04

(d) (c) (b) (a)

(lexp – lfit)lfit

0.02

0.00

–0.02

–0.04 0.00 Fig. 3.139

0.02 0.04 Coverage (ML)

0.06

Difference between best model fit and experimental intensity for Ag deposited on Ag(100) at 80 K. (a) Random condensation. (b) Capture zone one nearest neighbor. (c) Zone of one nearest and next-nearest neighbor. (d) Zone of two nearest-neighbor steps (after Félix et al. [218]).

One year after Vandoni et al., Gilmore and Sprague [219] did interesting simulations for deposition of Cu and Ag atoms on Ag(001), relying on embedded atom potentials as well as potentials by Haftel et al. [139]. In their work they found no evidence for transient diffusion of atoms condensing on a bare surface. However,

162

Atomic events in surface diffusion

interactions with adatoms and islands brought about atom spreading, conveying the impression of transient motion. A few years earlier Gilmore and Sprague [220] looked at the distribution of silver atoms impinging on a Ag(111) surface with energies between 0.1 and 10 eV. For low energies, ~0.1 eV, they did observe a random distribution of atoms on the surface, indicating no transient mobility. For higher energies, from 1 to 10 eV, ballistic displacements started occurring. These have also been observed by Villarba and Jónsson [221] for Pt on Pt(111). Ballistic exchange from Ir atoms colliding with Rh(100) was later found by Kellogg [213]. The introduction of the scanning tunneling microscope now led to a number of interesting and sometimes confusing studies which dealt with transient diffusion upon dissociation of diatomic molecules on a surface. We will discuss these later. Our main concern here is transient diffusion during adsorption of metal atoms, and we will for the moment continue this emphasis. In 1997 Hitzke et al. [216] used the STM to observe a random distribution of Ni atoms on the Au(110)-(1 × 2) reconstructed surface. They deposited 0.05 ML at 130 K and 0.2 ML at 180K, with immediate quenching to 130 K. Then they studied the lateral and vertical distribution of Ni atoms in the middle of the terrace and counted the number of monomers, dimers, trimers and linear chains as well as the spacing between them. They also did Monte Carlo simulations to compare with the experimental distribution. Based on this comparison they concluded that atoms could on average make no more than 0.5 hops during energy dissipation–energy exchange between the incoming atom and the surface was very efficient. The deposition of iron on Ag(100) was reported by Canepa et al. [222] in 1997, using thermal energy helium atom diffraction. Based on island size in the first layer they concluded that transient diffusion of one hop was consistent with their data. However, their data are a bit puzzling since they also found evidence for intermixing, from which they deduced exchange as the mechanism of movement over the surface. At the same time, Yue et al. [223,224] did molecular dynamics simulations of the low temperature growth of copper on the Cu(100) plane, relying on hybrid tight-binding-like potentials. They looked for transient diffusion as the amount of copper deposited on the surface was increased. A number of possible scenarios for creating the distributions shown in Fig. 3.140 was considered. These were 1 – direct deposition with lateral movement less than 1 spacing; 2 – direct deposit with lateral movement less than 2; 3 – downward funneling with a distance 0.35 to 0.7 lattice spacings; 4 – downward lattice funneling with a distance of 0.7 to 1 lattice spacing; 5 – downward funneling with a distance from 1 to 1.5 lattice spacings; 6 – impact cascade diffusion when atom takes the place of kicked atom; 7 – impact cascade diffusion when atom does not take the place of the kicked atom. As seen in Fig. 3.140, transient diffusion became significant only after a few atoms had been deposited. Their conclusion – “transient motion primarily stems from the impacting atoms interacting with other adatoms already on the surface” – confirmed the results previously obtained by Gilmore and Sprague [219]. At this stage it should be noted that whether or not transient diffusivity is important has little influence on experiments to measure diffusivities, as these are generally done on thermally equilibrated systems. The situation is different, however, for

163

3.4 Transient diffusion

1.00 0.75

80

T = 100 K

1 40

7 3

20

4

5

6

0

0.00 1.00

80

T = 200 K Number

Diffraction Intensity

0.25

0.50 0.25

T = 200 K

2

60 40

7

1 3

20

4 5

6

0

0.00 1.00 0.75

T = 100 K

60

0.50

0.75

2

T = 300 K

80

T = 300 K

2

60 0.50

7 40

0.25

20 0.00

Fig. 3.140

0 50 100 150 200 Number of atom deposited

1 3 4

6 5

0 Type of transient motions

Simulations of deposition of Cu atoms with 0.1 eVenergy on Cu(100). Left: Diffraction intensity as a function of the number of deposited atoms. Right: Number of different transient motions occurring during deposition. Transient diffusion occurs only after some atoms have been condensed (after Yue et al. [224]).

experiments in which the saturation island density is measured to learn about the diffusivity; here atoms are constantly colliding with the surface, and atoms excited during the collision process could produce elevated values for the diffusivity. This has been checked by Brune et al. [225] for the deposition of silver on Pt(111) at 35 K, at which temperature ordinary diffusion is effectively stopped. An STM image of the surface in Fig. 3.141a reveals individual spots, most of which correspond to atoms; the mean size of islands formed is plotted in Fig. 3.141b against the coverage. If transient mobility were active over one spacing, the mean island size would be 2.3 atoms at a coverage of 0.1 monolayer; the experiments, however, showed a size of 1.2±0.3 atoms, close to the value expected in the absence of any type of diffusion. Michely and Krug [226] also reported an STM survey of a Pt(111) surface at 23 K on which Pt atoms had been deposited. The number of image spots again corresponded to the number expected for a random distribution without any transient diffusion. In looking at all the work on transient diffusion of metal atoms it is important to note that interest in this topic was really stimulated by experiments indicating crystal growth at low temperatures. All the studies, including both experimental and theoretical efforts, that have examined the impact of individual metal atoms with a surface are in agreement – transient diffusion is a negligible phenomenon on an empty surface. The situation can be

164

Atomic events in surface diffusion

Fig. 3.141

Silver deposition on Pt(111) surface. (a) STM image of silver deposited on Pt(111) surface at 35 K shows predominantly atoms. (b) Condensation without any diffusion compared with transient diffusion over one site, “easy attachment.” Results indicate absence of transient motion (after Brune et al. [225]).

more complicated for weakly bound gases like Xe, which are likely to show some transient motion. Simulations have also suggested that some mobility may arise on a surface at a finite coverage in collisions or interactions between incoming and adsorbed atoms or island. The subject is therefore reasonably well understood.

3.4.2

Transient motion of dissociating molecules For the condensation of metal atoms on metal surfaces the conclusion from all the available experiments is that energy transfer between incoming atoms and the lattice is good so that transient diffusion does not play a significant role. Because of the great interest that has arisen in what happens during dissociation of diatomics on a surface we will, however, discuss this related topic at least briefly. Traditionally, on dissociation of a molecule on a surface, the atoms were supposed to end up in adjacent nearest-neighbor sites. Is this what really happens? It is conceivable that some of the energy released in the dissociation may translationally impel the atoms over the surface to positions far removed from each other. But it is also possible that the atoms start repelling each other after reaching a certain separation. The possibility of transient diffusion was first examined by Brune et al. [227,228], who looked with a scanning tunneling microscope at the distribution of oxygen atoms on Al(111) after adsorption of O2 at 300 K, a temperature at which oxygen atoms are immobile. After the chemisorption process, rather than atom pairs they found a random distribution of atoms on the surface, shown in Fig. 3.142, with an average separation between adsorbed atoms of 80 Å, suggesting the presence of translationally “hot” oxygen atoms. The histograms of the size distribution in Fig. 3.143 for several exposures of oxygen were also consistent with the random creation of islands from single atoms. This study started a number of investigations.

3.4 Transient diffusion

165

Fig. 3.142

STM images of O2 adsorption on Al(111) surface. (a) Clean surface. (b) After exposure to six monolayers of O2, black dots show atoms and small oxygen islands (after Brune et al. [228]).

Fig. 3.143

Distribution of oxygen island sizes, demonstrating increased collisions of adatoms with increasing coverage. No preference is indicated for pairs of adjacent adatoms (after Brune et al. [228]).

The findings by Brune et al. [227,228], of very large spacings of O adatoms in dissociation of O2 on aluminum led Engdahl and Wahnström [229] to do molecular dynamics simulations using an effective medium approximation for interactions. They looked at impacts with two different energies, 3.5 eV and 9.5 eV, with the direction of impact in both cases parallel to the surface. The average displacement between two atoms formed by dissociation was reported as 16.1 Å for an initial energy of 3.5 eV,

166

Atomic events in surface diffusion

Fig. 3.144

STM images of oxygen adsorption on Pt(111) at ~160 K. (a) Two monolayers of O2. (b) 1.2 monolayers of O2. Figures give distance between oxygen atoms (after Wintterlin et al. [231]).

considerably smaller than found in the experiments. For an initial energy of 9.5 eV the separation on dissociation was 52 Å, still lower than in the experiment. The separation in a completely random distribution of atoms should be equal to 35.6 Å for a coverage of 0.0014, which is consistent with that of Brune et al., so the value of 80 Å is a bit suspicious. The higher energy used in the simulations to come closer to the experimental findings of Brune et al. is likely to be too high compared to the experiments. In 1995, Jacobsen et al. [230] looked briefly at oxygen on Al(111) using densityfunctional theory calculations. They suggested that a cannon-like trajectory for oxygen atoms, in which an atom spends most of its time far from the surface, could possibly account for long distances between atoms in dissociation. A year later, Wintterlin et al. [231] studied the dissociation of O2 on the Pt(111) plane, using the STM to ascertain the distribution of oxygen atoms over the surface at 163 K; thermal motion of atoms was found around 200 K. An image of the surface with atoms on it is shown in Fig. 3.144. Pairs were found with quite a small inter-atomic separation, with oxygen occupying three fold fcc sites. A chart of the distribution giving a statistical account is shown in Fig. 3.145; the average distance between O atoms is roughly two lattice spacings. They also observed agglomeration of oxygen atoms at only 60 K, indicating mobility. The authors point out “that dissociation of O2 on Pt(111) actually creates ‘hot’ atoms … This leads, however, to only a very limited transient motion.” At the same time, Wahnström et al. [232] did dynamical simulations for adsorption of oxygen on Al(111) using a model potential fitted to first principle data. They did not find any evidence for the large transient mobility seen by Brune et al. [228]. In this investigations atoms were separated by 1 to 3 atomic spaces, results similar to what was found in experiments on the adsorption of oxygen on Pt(111) [231].

3.4 Transient diffusion

167

Fig. 3.145

Oxygen adsorption on Pt(111). (a) Model of Pt(111) with oxygen after dissociation. (b) Distribution of inter-pair distances between oxygen atoms (after Wintterlin et al. [231]).

Fig. 3.146

STM image at high resolution of Cu(110), showing both atomic and molecular oxygen (after Briner et al. [233]).

The study of oxygen adsorption continued with the work of Briner et al. [233], who now examined the adsorption of O2 on Cu(110) in a low-temperature STM. The oxygen impinged on the sample at an angle of only ~ 5° and with an energy near 50 meV. They found atom pairs at an inter-atomic spacing of 2 nearest neighbors at 4 K, usually aligned along the channels of the plane, as in Fig. 3.146. Precursor molecules were also seen at the surface but oriented across the channels. The distances they observed were roughly what Wintterlin et al. found on Pt(111) [231]. At the same time dissociation of O2 on Pt(111) at temperatures below 100 K was also studied by Stipe et al. [234] in several ways using the STM. When dissociation was initiated by photons, electrons, or by heating, the distance between two O adatoms ranged

168

Atomic events in surface diffusion

Fig. 3.147

Interaction of Cl2 with TiO2(110). (a) STM image of TiO2(110) surface, exposed to 0.07 Langmuir Cl2 at 300 K. Chlorine atoms indicated by white squares, Cl-Cl pairs by circle. (b) Model of TiO2(110) surface exposed to Cl2. Ti indicated by black spheres. A shows gas phase chlorine molecule, B and D show flat molecules, C gives Cl−. Upright Cl2 is labeled E (after Diebold et al. [235]).

from 1 to 3 lattice spacings, with the latter favored. The results are again in agreement with the previous studies by Wintterlin et al. [231] indicating a lack of transient mobility. In somewhat similar work, Diebold et al. [235] looked at the dissociation of Cl2 on TiO2(110). Although the substrate is not a metal, the findings are relevant here. The Cl atoms were observed in the STM and were found separated by an average distance of 26 Å, as indicated in Fig. 3.147a. Diebold et al. found only 23% pairs in the same row, 42% one row apart, and 25% two rows apart; 10% of the chlorine did not have a partner. To account for this, the authors propose that dissociation occurred from an upstanding molecule, in which the upper atom is emitted in a cannon-ball-like path, as suggested in Fig. 3.147b, rather than transient motion connected with the impact of the molecule. The cannon-ball mechanism of dissociation was already proposed by Jacobsen et al. [230] to account for the long distances seen by Brune et al. [227,228] between oxygen atoms on Al(111). On Ni(110), pairs of chlorine atoms were observed by Fishlock et al. [236] at a distance around 3.5 Å, aligned along the [001] direction, but it is not clear if transient motion influenced this distance. Dissociation of oxygen molecules on Rh(110) was examined with the STM by Hla et al. [237]. Oxygen atoms were found at a separation of 3.3 Å along the [001] direction, slightly smaller than the [001] lattice constant. Hla et al. concluded that atoms were sitting at adjacent asymmetric short bridge sites. Their findings did not support creation of “hot” atoms in molecular dissociation. The early studies of O2 dissociation on Al(111) were so startling that work on this system has continued, with rather surprising findings. Schmid et al. [238] in 2001 examined an aluminum (111) surface after O2 adsorption at different temperatures and found the images in Fig. 3.148a. Oxygen struck the sample at an angle of 60°. Schmid et al. measured the distance of 420 pairs on the surface, and pffiffiffifound that atom pairs were reasonably close to each other, separated by no more than 7 times the nearest-neighbor

169

3.4 Transient diffusion

0.6 as adsorbed

0.5 occurence

slightly annealed 0.4 0.3 0.2 0.1 0 1

√3

2 √7

3 2√3 √13

4 √19 √21

5

(b)

Fig. 3.148

Interaction of O2 with Al(111) surface. (a) Al(111) surface imaged with STM at 80 K, after O2 exposure at 150 K. Oxygen pairs are labeled with interatomic distance (in terms of Al nearestneighbor spacing ‘0 ). (b) Pair separations of oxygen atoms on Al(111) plane (after Schmid et al. [238]).

spacing of aluminum. A graph of the observed distribution in Fig. 3.148b shows the average inter-atomic distance as 5 Å. After annealing to ~ 250 K the number of nearestneighbor oxygen pairs increased, but there were also some larger separations observed. The behavior of O2 dissociation on Al(111) now appears very similar to oxygen dissociation on Pt(111) – transient mobility is low. This study did not show a preferential orientation of the pairs, and the question remains, does the direction of the incident beam matter? The behavior of O2 on the Al(111) plane is not yet clear. Komrowski et al. [239] looked at the surface with a scanning tunneling microscope, but also examined the gas phase with resonant enhanced multi-photon ionization. They dosed oxygen at an angle of 45° with the sample. On the surface they found single O atoms, O atoms at adjacent sites, or small islands of three atoms. The three atom islands were attributed to consecutive adsorption at adjacent surface sites, while the isolated single O adatoms were assumed due to abstractive chemisorption. The oxygen molecule sat on the surface in an end-on geometry, in which the upper atom is lost to the vacuum in dissociation. The important point, however, is that the ejected O atoms were detected in the gas phase. Abstractive chemisorption was proposed earlier for fluorine atom adsorption on Si(100)-(2 × 1) surface [240]. There this mechanism was explained by the presence of dangling bonds. The overall result of the more recent investigations is that atom pairs produced by dissociation of diatomic O2 are close together, and show little diffusion of atoms excited in the dissociation event. Recently, however, in 2001, Schintke et al. [241] have studied the dissociative adsorption of O2 on Ag(100). At a low concentration of 0.13% of a monolayer, with the surface at 50 K, they obtained the STM image in Fig. 3.149. This shows atom pairs at two kinds of separations: the smaller one has a separation of 20 Å (half of the oxygen pairs), the larger one 40 Å (one-third). The rest is unpaired, which might be a result of

170

Atomic events in surface diffusion

Fig. 3.149

STM images of oxygen adsorbed on Ag(100). (a) 0.13% of a monolayer on the surface. (b) 0.5% of a monolayer adsorbed. Pairs are indicated by white lines (after Schintke et al. [241]).

Fig. 3.150

Location of oxygen atoms produced by thermal dissociation on Pd(111) at 160 K. Inset shows molecules perpendicular and parallel to line from impurity, in white, to the center of the pair (after Rose et al. [242]).

abstractive dissociation with desorption of the second atom from the surface. These spacings are much larger than inter-atomic distances reported recently in dissociation, and the authors suggest two alternative mechanisms that may account for the observations. In the first, if the dissociation energy is released equally, the inter-atomic separation will be smaller than if the energy is distributed unequally for the two adatoms. An alternative mechanism attributes the smaller separation to dissociation from a molecular precursor, the longer separation to dissociation from the gas phase. So far the actual process responsible for the long distances has not been identified, but O2 on Ag(100) appears as the only example in which atoms from the dissociation are widely separated. This is especially surprising given that Ag(100) is an energetically rough surface.

References

Fig. 3.151

171

Oxygen atoms produced on Pd(111) by electron bombardmentpatffiffiffi 0.1 V bias and 10 nA current. (a) Pairs of oxygen atoms. Separation in higher resolution inset is 3 spacings along 511
24. (b) Pair distribution obtained in several experiments (after Rose et al. [242]).

In 2004, Rose et al. [242] looked at the dissociation of oxygen molecules on a Pd(111) surface under thermal conditions at 160 K. At this temperature dissociation occurs readily, pffiffibut ffi migration is not active yet. In most cases atoms were separated by distances of 3 or 2 times the lattice constant, as shown in Fig. 3.150, but the pair distribution was strongly influenced by the presence of impurities in the Pd crystal. Dissociation caused by injection of electrons from the tip at 100 mV and 10 nA was also examined. Rose et al. observed pairs at distances 1, √3, 2 or √7 times the lattice constant of Pd, as in Fig. 3.151. There was no indication of any significant transient motion in this system. It is clear that transient diffusion in molecular dissociation is a complicated, not terribly well understood event. It is difficult to separate effects associated only with dissociation from transient motion caused by impact with the surface. Dissociation of molecular oxygen generally leads to only brief diffusion of the dissociating atoms, something on the order of two spacings. For metal atoms, however, the situation is much more straightforward. The general impression is that transient diffusion is insignificant.

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[185] S. C. Wang, U. Kürpick, G. Ehrlich, Surface diffusion of compact and other clusters: Irx on Ir(111), Phys. Rev. Lett. 81 (1998) 4923–4926. [186] M. Schunack, T. R. Linderoth, F. Rosei, E. Laegsgaard, I. Stensgaard, F. Besenbacher, Long jumps in the surface diffusion of large molecules, Phys. Rev. Lett. 88 (2002) 156102 1–4. [187] M.-C. Marinica, C. Barreteau, D. Spanjaard, M.-C. Desjonque`res, Diffusion rates of Cu adatoms on Cu(111) in the presence of an adisland nucleated at fcc or hcp sites, Phys. Rev. B 72 (2005) 115402 1–16. [188] N. B. Cabrera, The structure of crystal surfaces, Discussions Faraday Soc. 28 (1959) 16–22. [189] R. W. Zwanzig, Collision of a gas atom with a cold surface, J. Chem. Phys. 32 (1960) 1173–1177. [190] G. Ehrlich, in: Metal Surfaces: Structure, Energetics and Kinetics, Adsorption and surface structure, (ASM, Metals Park, Ohio, 1963), p. 221–258. [191] G. Ehrlich, An atomic view of adsorption, Brit. J. Appl. Phys. 15 (1964) 349–364. [192] B. McCarroll, G. Ehrlich, Trapping and Energy Transfer in Atomic Collisions with a Crystal Surface, J. Chem. Phys. 38 (1963) 523–532. [193] G. Ehrlich, in: Proc. 9th Internat’l Vacuum Cong. and 5th Internat’l Conf. on Solid Surfaces, Invited Speakers’ Volume, J. L. deSegovia (ed.), Layer growth – an atomic picture, (ASEVA, Madrid, 1983), p. 3–16. [194] T. Gurney, F. Hutchinson, R. D. Young, Condensation of tungsten on tungsten in atomic detail: Observations with the field-ion microscope, J. Chem. Phys. 42 (1965) 3939–3942. [195] R. D. Young, D. C. Schubert, Condensation of tungsten on tungsten in atomic detail: Monte Carlo and statistical calculations vs experiment, J. Chem. Phys. 42 (1965) 3943–3950. [196] J. C. Tully, Stochastic-trajectory simulations of gas–surface interactions: Xe on Pt(111), Faraday Discuss. Chem. Soc. 80 (1985) 291–298. [197] M. Schneider, A. Rahman, I. Schuller, Role of relaxation in epitaxial growth: A moleculardynamics study, Phys. Rev. Lett. 55 (1985) 604–606. [198] D. E. Sanders, D. M. Halstead, A. E. DePristo, Metal/metal homoepitaxy on fcc(111) and fcc(001) surfaces: Deposition and scattering from small islands, J. Vac. Sci. Technol. A 10 (1992) 1986–1992. [199] H.-W. Fink, Mono-atomic tips for scanning tunneling microscopy, IBM J. Res. Develop. 30 (1986) 460–465. [200] H.-W. Fink, Point source for ions and electrons, Physica Scripta 38 (1988) 260–263. [201] W. F. Egelhoff, I. Jacob, Reflection high-energy electron diffraction (RHEED) oscillations at 77 K, Phys. Rev. Lett. 62 (1989) 921–924. [202] C. Koziol, G. Lilienkamp, E. Bauer, Intensity oscillations in reflection high-energy electron diffraction during molecular beam epitaxy of Ni on W(110), Appl. Phys. Lett. 51 (1987) 901–903. [203] D. K. Flynn-Sanders, J. W. Evans, P. A. Thiel, Homoepitaxial growth on Pd(100), Surf. Sci. 289 (1993) 75–84. [204] J. W. Evans, D. E. Sanders, P. A. Thiel, A. E. DePristo, Low-temperature epitaxial growth of thin metal films, Phys. Rev. B 41 (1990) 5410–5413. [205] D. E. Sanders, A. E. DePristo, Metal/metal homoepitaxy on fcc(001) surfaces: Is there transient mobility of adsorbed atoms? Surf. Sci. 254 (1991) 341–353. [206] P. S. Weiss, D. M. Eigler, Adsorption and accommodation of Xe on Pt{111}, Phys. Rev. Lett. 69 (1992) 2240–2243. [207] D. M. Eigler, E. K. Schweizer, Positioning single atoms with a scanning tunnelling microscope, Nature 344 (1990) 524–526.

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[208] H. J. Ernst, F. Fabre, J. Lapujoulade, Growth of Cu on Cu(100), Surf. Sci. 275 (1992) L682–684. [209] P. Bedrossian, B. Poelsema, G. Rosenfeld, L. C. Jorritsma, N. N. Lipkin, G. Comsa, Electron density contour smoothing for epitaxial Ag islands on Ag(100), Surf. Sci. 334 (1995) 1–9. [210] P. Blandin, C. Massobrio, Diffusion properties and collisional dynamics of Ag adatoms and dimers on Pt(111), Surf. Sci. 279 (1992) L219–224. [211] G. DeLorenzi, G. Ehrlich, Energy transfer in atom condensation on a crystal, Surf. Sci. Lett. 293 (1993) L900–907. [212] S. C. Wang, G. Ehrlich, Atom condensation at lattice steps and clusters, Phys. Rev. Lett. 71 (1993) 4174–4177. [213] G. L. Kellogg, Experimental observation of ballistic atom exchange on metal surfaces, Phys. Rev. Lett. 76 (1996) 98–101. [214] C.-L. Liu, J. B. Adams, Diffusion behavior of single adatoms near and at steps during growth of metallic thin films on Ni surfaces, Surf. Sci. 294 (1993) 197–210. [215] J. Jacobsen, K. W. Jacobsen, P. Stoltze, J. K. Nørskov, Island shape-induced transition from 2D to 3D growth for Pt/Pt(111), Phys. Rev. Lett. 74 (1995) 2295–2298. [216] A. Hitzke, M. B. Hugenschmidt, R. J. Behm, Low temperature Ni atom adsorption on the Au(110)-(1×2) surface, Surf. Sci. 389 (1997) 8–18. [217] G. Vandoni, C. Félix, R. Monot, J. Buttet, W. Harbich, Neighbor driven mobility of silver adatoms on Pd(100) measured by thermal helium scattering, Surf. Sci. 320 (1994) L63–67. [218] C. Félix, G. Vandoni, W. Harbich, J. Buttet, R. Monot, Surface mobility of Ag on Pd(100) measured by specular helium scattering, Phys. Rev. B 54 (1996) 17039–17050. [219] C. M. Gilmore, J. A. Sprague, A molecular dynamics study of transient processes during deposition on (001) metal surfaces, J. Vac. Sci. Technol. A 13 (1995) 1160–1164. [220] C. M. Gilmore, J. A. Sprague, Molecular-dynamics simulation of the energetic deposition of Ag thin films, Phys. Rev. B 44 (1991) 8950–8957. [221] M. Villarba, H. Jónsson, Atomic exchange processes in sputter deposition of Pt on Pt(111), Surf. Sci. 324 (1995) 35–46. [222] M. Canepa, S. Terreni, P. Cantini, A. Campora, L. Mattera, Initial growth morphology in a heteroepitaxial system at low temperatures: Fe on Ag(100), Phys. Rev. B 56 (1997) 4233–4242. [223] Y. Yue, Y. K. Ho, Z. Y. Pan, Low-temperature epitaxy and transient diffusion mechanisms on Cu(100), Europhys. Lett. 40 (1997) 453–457. [224] Y. Yue, Y. K. Ho, Z. Y. Pan, Molecular-dynamics study of transient-diffusion mechanisms in low-temperature epitaxial growth, Phys. Rev. B 57 (1998) 6685–6688. [225] H. Brune, G. S. Bales, H. Jacobsen, C. Boragno, K. Kern, Measuring surface diffusion from nucleation island densities, Phys. Rev. B 60 (1999) 5991–6006. [226] T. Michely, J. Krug, in: Islands, Mounds and Atoms, (Springer-Verlag, Berlin, 2004), Section 2.1. [227] H. Brune, J. Wintterlin, R. J. Behm, G. Ertl, Surface migration of “hot” adatoms in the course of dissociative chemisorption of oxygen on Al(111), Phys. Rev. Lett. 68 (1992) 624–626. [228] H. Brune, J. Wintterlin, J. Trost, G. Ertl, J. Wiechers, R. J. Behm, Interaction of oxygen with Al(111) studied by scanning tunneling microscopy, J. Chem. Phys. 99 (1993) 2128–2148. [229] C. Engdahl, G. Wahnström, Transient hyperthermal diffusion following dissociative chemisorption: a molecular dynamics study, Surf. Sci. 312 (1994) 429–440. [230] J. Jacobsen, B. Hammer, K. W. Jacobsen, J. K. Nørskov, Electronic structure, total energies, and STM images of clean and oxygen-covered Al(111), Phys. Rev. B 52 (1995) 14954–14962.

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[231] J. Wintterlin, R. Schuster, G. Ertl, Existence of a “hot” atom mechanism for the dissociation of O2 on Pt(111), Phys. Rev. Lett. 77 (1996) 123–126. [232] G. Wahnström, A. B. Lee, J. Strömquist, Motion of “hot” oxygen adatoms on corrugated metal surfaces, J. Chem. Phys. 105 (1996) 326–336. [233] B. G. Briner, M. Doering, H.-P. Rust, A. M. Bradshaw, Mobility and trapping of molecules during oxygen adsorption on Cu(110), Phys. Rev. Lett. 78 (1997) 1516–1519. [234] B. C. Stipe, M. A. Rezaei, W. Ho, Atomistic studies of O2 dissociation on Pt(111) induced by photons, electrons, and by heating, J. Chem. Phys. 107 (1997) 6443–6447. [235] U. Diebold, W. Hebenstreit, G. Leonardelli, M. Schmid, P. Varga, High transient mobility of chlorine on TiO2(110): Evidence for “cannon-ball” trajectories of hot adsorbates, Phys. Rev. Lett. 81 (1998) 405–408. [236] T. W. Fishlock, J. B. Pethica, F. H. Jones, R. G. Egdell, J. S. Foord, Interaction of chlorine with nickel (110) studied by scanning tunnelling microscopy, Surf. Sci. 377–379 (1997) 629–633. [237] S. W. Hla, P. Lacovig, G. Comelli, A. Baraldi, M. Kiskinova, R. Rosei, Orientational anisotropy in oxygen dissociation on Rh(110), Phys. Rev. B 60 (1999) 7800–7803. [238] M. Schmid, G. Leonardelli, R. Tscheliesznig, A. Biedermann, P. Varga, Oxygen adsorption on Al(111): low transient mobility, Surf. Sci. 478 (2001) L355–362. [239] A. J. Komrowski, J. Z. Sexton, A. C. Kummel, M. Binetti, O. Weisze, E. Hasselbrink, Oxygen abstraction from dioxygen on the Al(111) surface, Phys. Rev. Lett. 87 (2001) 246103 1–4. [240] Y. L. Li, D. P. Pullman, J. J. Yang, A. A. Tsekouras, D. B. Gosalvez, K. B. Laughlin, Z. Zhang, M. T. Schulberg, D. J. Gladstone, M. McGonigal, S. T. Ceyer, Experimental verification of a new mechanism for dissociative chemisorption: Atom abstraction, Phys. Rev. Lett. 74 (1995) 2603–2606. [241] S. Schintke, S. Messerli, K. Morgenstern, J. Nieminen, W.-D. Schneider, Far-ranged transient motion of “hot” oxygen atoms upon dissociation, J. Chem. Phys. 114 (2001) 4206–4209. [242] M. K. Rose, A. Borg, J. C. Dunphy, T. Mitsui, D. F. Ogletree, M. Salmeron, Chemisorption of atomic oxygen on Pd(111) studied by STM, Surf. Sci. 561 (2004) 69–78.

4

Diffusion on one-dimensional surfaces

Quite a number of diffusivity measurements have been reported during the last 40 years, not all of them of the same quality. Our aim here will be to outline the data currently existing for diffusivities, which allow us to describe diffusion of single atoms on metal surfaces. The quantity that needs to be known in order to calculate the flux of atoms diffusing over the surface is the diffusivity D, which establishes the relation between the flux J and the concentration gradient ∂c=∂x, according to Eq. (1.1) J = −D∂c=∂x. The diffusivity has to be known at different temperatures, and is therefore written as Eq. (1.23),   DED , where Do, the diffusivity prefactor, is usually considered a conD ¼ Do exp
kT stant that can be related to the frequency prefactor by Eq. (1.22) and (1.23), Do ¼ 0 ‘2 ; here ‘2 is the mean-square jump length, and 0 is the prefactor for atomic jumps. Our emphasis in this chapter will be largely upon the activation energy for diffusion, which from now on we write more simply as ED instead of DED , and on the prefactor Do. Many systems have been examined by a variety of techniques, and with different degrees of reliability. What we will therefore do is to outline the facts reported for diffusivities of single atoms in the order in which the substrate appears in the periodic table. We will also comment on the limitations of the techniques used and the advantages of the approach employed. The presentation is divided into two parts: diffusivities on one-dimensional, channeled surfaces are treated first; only then will we examine the data on two-dimensional surfaces. For channeled surfaces we avoid some of the complications that arise on two-dimensional planes, and diffusion is therefore easier to discuss. We will not touch mass transport phenomena of many atoms at all, nor will we be concerned here in detail with the mechanism of diffusion or with atomic jump processes; that has already been discussed at length in the previous chapter. Just the empirical description of the diffusivities, and sometimes a comparison of different results will be attempted. We will not examine all studies; we tend to downplay measurements at but a single temperature, from which the diffusivity is obtained by assuming a value for the prefactor Do. The aim of diffusion studies is the assignment of values for both the prefactor and the diffusion barrier, because at the moment these are the quantities that are generally difficult to calculate reliably. In the survey that follows, the emphasis is on experimental results, and measurements of diffusivities by a variety of different techniques are presented. However, we must note a caution here. Most, but not all, scanning tunneling microscopy studies have measured

184

Diffusion on one-dimensional surfaces

the saturation island density to arrive at the diffusivity. There are several advantages to this approach, but also a big defect: this approach relies on the predictions of nucleation theory, which are based on the assumption that long-range atomic interactions can be ignored. This approximation is known to be incorrect at least for some systems, and may well vitiate results. In recent years theoretical estimates of diffusivities have become popular, and have been attempted using different approaches. The results from these efforts will be presented in conjunction with experiments, but perhaps not as carefully as the experimental studies. In order to keep the presentation simple, we will, as already pointed out, start our survey by focusing on diffusion on one-dimensional, that is channeled planes. In examining the work done to characterize atom diffusion on such surfaces, it will be important to keep in mind the geometry of the planes discussed. To make this task easier we have at the very start of the previous chapter shown hard-sphere models of the important channeled planes – the bcc(211) and (321) in Fig. 3.1, as well as fcc(110), (311), and (331) in Fig. 3.2. Their atomic arrangements are similar, but nevertheless their properties are quite different.

4.1

Aluminum: Al(110), (311), (331) The first measurements of self-diffusion on aluminum were done by Tung [1] in 1981, who looked at (110), (311), and (331) planes in the field ion microscope. He determined the temperatures for the onset of diffusion in only ~20 trials, and by assuming a prefactor for the diffusivity of 10−3 cm2/sec was able to arrive at the following diffusion barriers, all in eV: (110)|| = 0.43, (110)? = 0.43, (311) = 0.48, (331) = 0.46. These three planes are all made up of channels, and for (311) and (331) diffusion took place along the channels. On the (110) plane, however, diffusion was two-dimensional, and occurred at roughly the same rate in the direction of the channels and transverse to them. The cross-channel movement on this plane is probably an indication of atom exchange as the diffusion mechanism for this system. Till today it is the one and only attempt to experimentally investigate single atom surface diffusion for this system. Even if the statistics are not great, the measurements are very valuable. Liu et al. [2] began calculational efforts by evaluating the self-diffusion barriers on several planes of aluminum, relying on embedded atom, EAM, potentials.1 On the (110) plane, the activation energy for jumps along the channels was found to be 0.26 eV with a diffusivity prefactor Do of 1.8 × 10−3 cm2/sec using AFW parameters [3], while for crosschannel processes the barrier proved to be 0.30 eV with a prefactor of 6.0 × 10−2 cm2/sec. In the VC approximation [4], cross-channel moves required an activation energy ED of only 0.15 eV with a prefactor 2.4 × 10−2 cm2/sec and, from this approach, exchange seems to be the likely mechanism for diffusion on this plane. However, AFW parametrization gave the opposite indication – hopping was more favorable, leaving the mechanism of movement uncertain for the Al(110) plane. On the (311) plane the diffusion barrier for 1

The abbreviations commonly used in theoretical evaluations are briefly described in Section 2.8.

185

4.1 Aluminum: Al(110), (311), (331)

Fig. 4.1

Trajectories of aluminum atom for diffusion on Al(110) surface calculated using effective medium potentials (after Gravil and Holloway [5]).

E = 0.0 eV Fig. 4.2

Bridge

Exchange

E = 0.204 eV

E = 0.285 eV

Bridge

E = 0.204 eV

E = 0.0 eV

Models for cross-channel motion by Al atom exchange on Al(110). Barrier heights arrived at with EMT potentials (after Gravil and Holloway [5]).

AFW parameters was 0.20 eV with a prefactor 2.0 × 10−3 cm2/sec; with VC potentials the activation energy was 0.24 eV and the prefactor turned out to be 6.7 × 10−3 cm2/sec. On (331) the barrier was 0.27 eV, with a prefactor of 2.0 × 10−3 cm2/sec (AFW potentials), and 0.24 eV and 5.4 × 10−3 cm2/sec (VC potentials). Only hopping was considered on these planes, however, it is the most likely mechanism and experimental findings seem to confirm that. Effective medium theory was used by Gravil and Holloway [5] in molecular dynamics simulations, relying on a realistic many-body potential for aluminum diffusion on Al(110). They observed hopping along the channels with an activation energy of 0.204 eV, as well as exchange processes with a higher barrier, 0.285 eV, as indicated in Figs 4.1 and 4.2; the prefactor for the diffusivity was not calculated, but again hopping seems to be slightly more favorable. Stumpf and Scheffler [6] did density-functional theory estimates for self-diffusion on Al(110). Jumps along the channels occurred over a barrier of 0.33 eV, while cross-channel diffusion over a much higher activation energy of 0.62 eV proceeded by atom exchange. Again the prefactor for diffusivity was not derived. Agrawal et al. [7] relied on LennardJones potentials for their work, which gave them a rather too big self-diffusion barrier of 0.67 eV with a prefactor of 3.5 × 10−3 cm2/sec for in-channel transitions. Cross-channel transitions, by jumps not exchange, occurred over a much higher barrier of 1.19 eV with a prefactor of 6.9 × 10−3 cm2/sec. A year later, calculations for self-diffusion of aluminum on Al(110) were also done by Sun and Li [8], relying on the Vienna ab-initio local-densityfunctional theory simulation package [9–11]. They considered three mechanisms of motion: in-channel jumps, cross-channel jumps, and atom exchange, with activation energies of 0.38 eV, 0.83 eV, and 0.50 eV respectively. No information was given for the prefactor of the diffusivity. From the theoretical results presented it appears that diffusion should proceed along channels, and atom exchange would only become more significant at

186

Diffusion on one-dimensional surfaces

elevated temperatures. However, this prediction seems to be contradicted by experiments where cross-channel motion was clearly observed [1]. The situation with aluminum appears to be confused: for the (110) plane, activation energies derived in experiments are unreliable due to limited statistics and do not agree with calculations with the exception of the work by Sun and Li [8], but calculations are not too close together either. What is worth noting, however, is that direct observation of cross-channel motion in the FIM provided proof of exchange on this surface. For (331) and (311) planes experiments are not in accord with the calculated energetics either, but it is clear that movement is one-dimensional on these surfaces, suggesting that diffusion most likely proceeds by in-channel hopping.

4.2

Nickel: Ni(110) Nickel is a comparatively soft metal, which field evaporates under the conditions for imaging with helium gas and makes FIM investigation difficult. However, because of its simple band structure and its technological significance, Tung and Graham [12] studied self-diffusion of nickel in 1980 on Ni(110) and (311) as well as (331) planes using the FIM. Imaging was done with neon, and after careful preparation to reduce defects it became possible to image adsorbed atoms. Arrhenius plots for diffusion were obtained with over 30 observations at each temperature and are shown in Fig 4.3. The results are unusual, especially on the (110) plane. As was already indicated in Fig. 3.46, rather different behavior was found on thermally treated samples and those heated in hydrogen and then field evaporated. The authors favored the former. On the (110) plane after thermal treatment the activation energy for self-diffusion was 0.23 ± 0.04 eV along the channels, but with a very suspicious prefactor for the diffusivity of 1 × 10−9 cm2/sec. Cross-channel motion also occurred, with a slightly higher barrier of 0.32 ± 0.05 eV and with a low prefactor Do = 10−7 cm2/sec. The low prefactors have raised concerns, and the data for Ni(110) have been reworked by Liu et al. [2], who presumed a prefactor of 10−3 cm2/sec. This yielded activation

Diffusivity D(cm2/sec)

Temperature T(K) 200 180 160 140 10–14

10–16

10–18

Fig. 4.3

(311) (331) (110)II

60

70 80 1/kT (eV–1)

90

Arrhenius plot for self-diffusion on Ni planes examined by field ion microscopy. Motion shown on (110) plane is along the channels (after Tung and Graham [12]).

187

4.2 Nickel: Ni(110)

energies of 0.45 eV both along the channels as well as across them, erasing the difference of 0.09 eV Tung and Graham reported between in- and cross-channel movement on the (110) plane. It should be noted, however, that the results on Ni(110) after hydrogen treatment look rather more reasonable, although a bit on the high side. The activation energy for in-channel motion was 0.30 ± 0.06 eV with Do = 101 cm2/sec, whereas for cross-channel diffusion the activation energy was lower, 0.26 ± 0.06 eV, with Do = 10−1 cm2/sec. In 1991 and 1992 Kellogg [13,14] reported on the diffusion of platinum adatoms over Ni(110) and (311) planes. The surfaces were treated by a combination of thermal cleaning, sputtering and field evaporation, but not hydrogen. For Ni(110), he obtained an estimate for the activation energy of 0.28 eV from the temperature for the onset of diffusion, assuming Do = 1 × 10−3 cm2/sec. From monitoring the voltage required for adatom field evaporation it was obvious that diffusion occurred by platinum atoms exchanging with nickel in the sides of the channels, as was shown in Fig. 3.50. Scattering of helium atoms from a Ni(110) surface was done by Graham et al. [15,16] in 1997, who measured the broadening of the energy distribution of helium atoms scattered from diffusing nickel. Measurements were done at temperatures from 900 K to 1200 K, and yielded an activation energy of 0.536 ± 0.040 eV. This figure was for atoms jumping along the rows, with 80% single jumps and 20% double jumps. For crosschannel diffusion they deduced a lower barrier of 0.424 ± 0.040 eV, as shown in Fig. 4.4. Since this value is not too far away it suggests coexistence of both mechanisms, in agreement with FIM findings. It must be emphasized that the barriers here are slightly higher than what has been found from previous FIM measurements at lower temperature. This may not be due to the presence of double jumps – it could just be for single jump movement of individual adatoms influenced by interactions. The technique is not capable of controlling atom distances directly. 200 150

ΔK = 0.43 Å–1 [110]

ΔE (μeV)

100

Eeff = 0.536 eV

50 30 20

10

[100]

Eeff = 0.424 eV 5

Fig. 4.4

8

9

11 10 1000/T (K–1)

12

13

Arrhenius plot for diffusion of Ni on Ni(110) derived from quasi-elastic He atom scattering. Note that cross-channel diffusion along [100] has a lower activation energy than in-channel motion (after Graham et al. [15]).

188

Diffusion on one-dimensional surfaces

T = 228 K, Ff = 1.5 × 10–4 ML/sec T = 270 K, Ff = 2.6 × 10–3 ML/sec T = 290 K, Ff = 2.6 × 10–3 ML/sec

Fig. 4.5

STM images of 0.09 ML nickel deposit on Ni(110), under different deposition conditions, showing narrow deposits along channels. Insets reveal island width, which at low temperature is monatomic, but wider at 290 K (after Memmel et al. [17]).

Fig. 4.6

Dependence of nickel island density on Ni(110) as a function of inverse temperature, yielding a diffusion barrier of 0.060 eV (after Memmel et al. [17]).

In 2005, scanning tunneling microscopy observations of the growth of nickel films on Ni(110) were made by Memmel et al. [17] over a number of temperatures for coverages up to 0.1 ML. Strongly elongated islands, shown in Fig. 4.5, were observed, revealing the anisotropy of the system. They found that despite the channels on the Ni (110) plane, nucleation was quasi-isotropic with a critical cluster size of one. From the temperature dependence of the island density shown in Fig. 4.6, they arrived at a diffusion barrier of 0.60 eV, in reasonable accord with the work of Graham [15] but not with single atom FIM findings, which indicates that barriers might be influenced by long-range interactions. It is important to note that a slight anisotropy of diffusion on Ni (110) was already observed in 1967 by Maiya and Blakely [18] who found a very small difference between

189

4.2 Nickel: Ni(110)

4.0

Diffusion in [001]direction

log D (D in cm2/sec)

Diffusion in [110]direction

6.0

7.0 5.5

Fig. 4.7

ED = 1.74 ± 0.04 eV Do = 12.8 ± 4 cm2/sec

5.0

ED = 1.85 ± 0.04 eV Do = 23.9 ± 6.4 cm2/sec

6.0

6.5

7.0 7.5 104/T (K–1)

8.0

8.5

9.0

Arrhenius plot for massive self-diffusion on Ni(110) plane, showing both in-channel and cross-channel motion (after Maiya and Blakely [18]).

movement along compared with movement in , shown in Fig. 4.7. This movement is not single atom diffusion, but the almost isotropic values of the activation energy were probably the first indication of atom exchange on this plane. Early diffusion studies were also done by Bonzel and Latta [19], who in 1978 carried out mass transfer experiments measuring the decay of a sinusoidal profile and compared these with theoretical investigations based on pairwise interactions. These also are not single atom studies, but the results, in Fig. 4.8 are interesting. At temperatures below 1000 K there appear to be two modes of diffusion, one along the direction, that is along the channels, over a barrier of 0.76 eV, and a much higher activation energy process of 1.95 eVacross them. Even if this value cannot be compared with single atom diffusion, as it is affected by interactions, these are among the early measurements showing anisotropic movement on this channeled surface of nickel. Theoretical studies on nickel surfaces started in 1980 with the work of Flahive and Graham [20] who investigated Ni as well as other fcc and bcc metals using Morse potentials. Their findings are summarized in Fig. 4.9a and b, and will also be useful for other surfaces. However, it is worth noting that Morse potentials ignore many-body interactions, which is probably the reason for the poor agreement of this data with later studies – values calculated this way are definitely inaccurate. Theoretical estimates with many-body potentials using EAM interactions for selfdiffusion on nickel surfaces were first done by Liu et al. [2,21] in 1991. On Ni(110), jumps along the channels occurred over a barrier of 0.44 eV for AFW potentials, with a prefactor of 2.3 × 10−3 cm2/sec; with VC potentials the activation energy was lower, 0.39 eV with a prefactor Do = 4.0 × 10−3 cm2/sec. For cross-channel motion the barriers were higher, 0.49 eV with a prefactor of 3.7 × 10−3 cm2/sec (AFW), and 0.42 eV with the prefactor 2.8 × 10−2 cm2/sec (VC). They also compared their results with investigations by Lennard-Jones and Morse potentials, and found no agreement; many-body effects

190

Diffusion on one-dimensional surfaces

10–3

1573 1373

Temperature T(K) 1173 973 873

773

Ni (110) [001] direction

Diffusivity D (cm2/sec)

10–4

[110] direction

10–5 ED = 0.76 eV 10–6

ED = 1.95 eV 10–7

10–8

Fig. 4.8

6

7

8

9 10 11 10000/T (K–1)

12

13

Arrhenius plot for diffusivities of Ni on Ni(110) for in-channel [1
10] and cross-channel [001] motion, measured in mass-transfer experiments. Strong anisotropy is clear at lower temperatures (after Bonzel and Latta [19]).

clearly play an important role on Ni surfaces. In 1993 Liu and Adams [22] investigated diffusion over steps on nickel using EAM potentials; for movement on the flat Ni(110) surface they once more obtained a value similar to their previous work: 0.41 eV for in-channel and 0.49 eV for cross-channel diffusion. In 1992 Rice et al. [23] did calculations for self-diffusion on nickel surfaces using EAM with four different interatomic potentials: Foiles, Baskes, and Daw [24] (FBD), Voter and Chen [4] (VC), Oh and Johnson [25] (OJ), and Ackland, Tichy, Vitek and Finnis [26] (ATVF). The potentials differ mostly in the interpretation of the electron density surrounding the atom and the embedding function. The VC approach is similar to FBD but pairwise interactions are given by Morse potentials. Estimates were made over a huge range of temperatures, 50–1600 K. For in-channel motion on Ni(110) they obtained the following values of the activation energy, in eV: 0.50 (FBD), 0.53 (VC), 0.44 (OJ) and 0.64 (ATVF). The spread in values is ~0.2 eV, about 40% of the barrier. The values obtained from VC potentials do not agree with the previous findings of Liu [2]: 0.53 eV compared with 0.39 eV for VC potentials and 0.50 eV versus 0.44 eV with AFW parameters. Prefactors from different potentials are in quite reasonable agreement, all are in the range of standard values: 3.3 (FBD), 3.8 (VC), 3.6 (OJ), 4.5 (ATVF), all multiplied by 10−3 cm2/sec. Rice et al. also investigated cross-channel movement, but only by hopping, not exchange; it

4.2 Nickel: Ni(110)

Fig. 4.9

191

Activation energies in self-diffusion of different single metal atoms, calculated using Morse potentials with complete substrate relaxation. (a) Fcc transition metals. (b) Bcc transition metals (after Flahive and Graham [20]).

required a much higher energy, with the following values, all in eV: 1.30 (FBD), 1.67 (VC), 1.01 (OJ), and 1.53 (ATVF). The diffusivity prefactors for hopping across the channels are slightly higher but in the same range of magnitude as for in-channel motion: 6.5 × 10−3 cm2/sec (FBD), 7.5 × 10−3 cm2/sec (VC), 6.5 × 10−3 cm2/sec (OJ), and 7.7 × 10−3 cm2/sec (ATVF). The spread of the Arrhenius plots for the different potentials is presented in Fig. 4.10. The authors did not consider exchange in their investigations, despite the clear indication of this mechanism from the experiments of Tung and Graham [12] as well as from the calculations of Liu et al. [2]. In 1994, Stoltze [27] examined self-diffusion on Ni(110) relying on EMT potentials. He estimated a barrier of 0.407 eV for in-channel jumps and 0.564 eV for cross-channel exchange while cross-channel jumps cost 1.157 eV. Again in-channel jumps seem to be the most favorable mechanism based on this study. The same year Perkins and DePristo [28] published calculations using MD/MC-CEM potentials of the activation energy for in-channel hopping as well as for atom exchange. For self-diffusion the activation energy for jumping decreased from 0.273 eV for 13 active atoms to 0.243 eV for 113 active atoms. For atom exchange similar changes were found – the barrier diminished from 0.427 eV to 0.392eV, showing that the size of the supercell used for the theoretical description of the surface influenced the results obtained for the activation energy. From CEM they obtained values of 0.18 eV for the hopping barrier and 0.35 eV for exchange. Six years later, Haug and Jenkins [29] investigated diffusion of Ni on the Ni (110) surface, working with a slab of 146 active atoms and using the EAM5 potential of Wonchoba et al. [30]. For isolated Ni atoms they came out with a barrier of 0.39 eV for hopping and 0.42 eV for exchange. What is more surprising is that the presence of hydrogen lowered the energy to 0.38 eV for both hopping and exchange. Hydrogen

192

Diffusion on one-dimensional surfaces

500

Temperature T(K) 250 167

125

0 (100) –40 –80

In D

–10 (110)[110] –30 –50 0 (110)[001]

–40 –80 –120 –160 2

Fig. 4.10

4 1000/T (K–1)

6

8

Arrhenius plots for self-diffusion on Ni(100) and Ni(110) calculated with different EAM potentials. Solid line – FBD; long-dashed – VC; short-dashed – OJ; dot-dashed – ATVF (after Rice et al. [23]).

seemed to influence the activation energy for exchange more than for hopping, making both mechanisms equally probable. The authors associated this findings with weak Ni-H bonding, which stabilized the transition state for Ni hopping. The time constants were 1.4 × 10−5 sec for hopping and 5.6 × 10−5 sec for exchange. More extensive calculations using EAM potentials to understand the physical meaning of prefactors for atom hopping were done by Kürpick [31]. For Ni(110) she found an activation energy of 0.39 eV with a prefactor (1.4 ± 0.1) × 10−3 cm2/sec for in-channel jumps, agreeing with prior studies. Her investigation concluded that the usual prefactor for the diffusivity should be observed for single atom movement. Agrawal et al. [7] relied on Lennard-Jones interactions to come up with an activation energy of 0.86 eV and a prefactor of 2.5 × 10−3 cm2/sec for atom jumps along channels. They estimated a barrier of 1.68 eV for cross-channel jumps with a prefactor of 4.0 × 10−2 cm2/sec. The barrier given here for in-channel jumps is clearly much too high, but the EAM estimates agree not too badly with the FIM measurements. Ndongmouo and Hontinfinde [32] did calculations based on RGL potentials and found a barrier of 0.32 eV for in-channel jumping, and a somewhat larger value, 0.38 eV, for cross-channel exchange processes. Cross-channel jumps required overcoming a much higher activation energy of 1.33 eV. Kong and Lewis [33] in 2006 did EAM calculations of the self-diffusion prefactor, summing up all vibrational contributions. For motion along the channels making up the plane they found a temperature independent value of 4.0 × 10−3 cm2/sec for atom hops, twice higher than one obtained from the local

4.3 Nickel: Ni(311) and Ni(331)

193

approximation [31], however, in the same magnitude range. Recently, Kim et al. [34] have done molecular dynamics studies of diffusion using RGL interactions. The barrier for in-channel jumps amounted to 0.301 eV, and the same value was found for crosschannel exchange suggesting coexistence of both mechanisms. Cross-channel jumps required a much higher energy of 1.026 eV. For a Cu adatom carrying out in-channel diffusion the barrier was 0.334 eV, for Pd 0.426 eV, for Ag 0.334 eV, for Pt 0.308 eV, and for Au 0.440 eV. The authors did not discuss their findings in detail. For Pt adatoms on Ni (110) Kim did not consider exchange as a mechanism, even though it was shown in experiments [13,14] that exchange not hopping was a leading mechanism. The calculated results for diffusion on Ni(110) are in reasonable agreement with the somewhat uncertain experimental findings, but clearly, more reliable experiments are desirable. However, the existence of the exchange mechanism for diffusion on Ni(110) seems to be documented. Even though it appears to be slightly more energetically demanding, both processes are clearly present in experiments. There is still a huge lack of data for hetero-diffusion on this plane, the only experiments having been done by Kellogg.

4.3

Nickel: Ni(311) and Ni(331) Compared with Ni(110), other fcc channeled planes, such as (331) and (311) have gained much less attention. The first investigation of these planes of nickel was done by Tung and Graham in 1978 [35] and in 1980 [12]. Arrhenius plots for diffusion on (311) and (331) planes were shown in Fig 4.3 but are based on only ~ 30 observations per temperature. Self-diffusion on the (311) plane took place over a barrier of 0.30 ± 0.03 eV, again with a low prefactor of 1.9 × 10−6 cm2/sec. The barrier to diffusion on (331) was higher, 0.45 ± 0.03 eV with a prefactor of 2.3 × 10−3 cm2/sec. For both these planes the movement of atoms is one-dimensional, an indication that movement is by hopping rather than by exchange with a lattice atom. Doubting the physical meaning of the low prefactor, Liu et al. [2] recalculated the data for these planes presuming a prefactor of 10−3 cm2/sec. On the (311) plane, the recalculated barrier amounted to 0.37 eV, compared to 0.30 eV in the original study; the value for the (331) plane stayed at 0.45 eV after recalculation. In 1992 Kellogg [14] reported on the diffusion of platinum adatoms over the Ni(311) plane. Diffusion observed at three temperatures took place by in-channel movement, over a barrier of 0.38 eV, estimated by assuming a value for the prefactor. As expected no exchange was observed on this surface. Theoretical studies on this surfaces started as well in 1980 with the work of Flahive and Graham [20] using Morse potentials. Their findings, summarized in Fig. 4.9a and b, are definitely too low, as their calculations considered only pair interactions. Many-body theoretical estimates using EAM interactions were done by Liu et al. [2,21]. Diffusion on (311) took place along the channels with an activation energy of 0.34 eVand a prefactor of 1.4 × 10−3 cm2/sec for AFW potentials, and a barrier of 0.38 eV and a prefactor of 4.4 × 10−3 cm2/sec using VC potentials. On the (331) plane the barriers were 0.45 eV and 0.46 eV for AFW and VC potentials, and the prefactors amounted to 1.4 × 10−3 and 4.2 × 10−3 cm2/sec. The comparison with investigations by Lennard-Jones

194

Diffusion on one-dimensional surfaces

and Morse potentials gave no agreement, indicating that many-body effects clearly play an important role on Ni surfaces. Merikoski et al. [36] in 1997 carried out EMT investigations for diffusion on the Ni(311) plane. The activation energy for in-channel hopping was 0.351 eV, very close to previous findings. The barrier to going over the edge by atom exchange at a straight step was more favorable than just hopping over the edge: 1.302 eV compared to 1.373 eV. Both from experiments and theoretical investigations it is clear that movement on these planes proceeds by hopping, and the diffusion parameters seem to be in quite reasonable agreement. However, except for one experiment, there is no information about hetero-diffusion on these planes.

4.4

Copper: Cu(110), Cu(311), Cu(331)

4.4.1

Self-diffusion The only experiment which derived values for the self-diffusion of copper on Cu(110) at small coverages was done by Robinson et al. [37] in 1996. Copper island growth on Cu(110) was measured in quasi-elastic helium atom scattering for a coverage of 0.5 ML, at which interactions are, however, very likely to influence the data. From the temperature dependence they were able to arrive at an activation energy of 0.84 ± 0.04 eV for diffusion across the channels on this plane, a value much higher than expected. They also were able to derive in the same way a value for in-channel movement, but since this turned out to be higher than for cross-channel motion, they disregarded it, based on number of bonds broken, as unphysical. Today we know that counting bonds is not a reliable method for diffusion estimates, and the lower value for cross-channel movement might be correlated with the existence of the exchange mechanism on this plane. Unfortunately, copper surfaces are not easily imaged in the FIM, but a considerable amount of work has been done on theoretical calculations for self-diffusion on copper. In fact, copper surfaces have received the most intense theoretical attention among metals. It is interesting that the main theoretical efforts have turned in a direction where there is essentially no experimental data available for comparison. Calculational work seems to have been started by Wynblatt and Gjostein [38], who depended on Morse potentials to represent atomic interactions. For in-channel motion on Cu(110) they calculated a very small barrier height of 0.059 eV. In 1980 Flahive and Graham [20] also did Morse calculations for copper; their data are shown in Fig. 4.9. Their value for the diffusion barrier on Cu(110) is also very small, but slightly higher than the barriers obtained for Cu(311) and Cu(331). Such small values are presumably an indication of the importance of many-body interactions. Liu et al. [2] used EAM interactions. With AFW parameters their barrier for in-channel motion was 0.23 eV, with a prefactor of 8 × 10−4 cm2/sec; with VC parameters their results were 0.28 eV for the barrier and 4.4 × 10−3 cm2/sec for the prefactor. In cross-channel motion by atom exchange they came up with a slightly higher barrier, 0.30 eV, with AFW potentials; the prefactor was 3.2 × 10−2 cm2/sec. With VC interactions they estimated a barrier of 0.31 eV for atom exchange with a prefactor of 2.7 × 10−2 cm2/sec. Surprisingly this

4.4 Copper: Cu(110), Cu(311), Cu(331)

195

very first many-body calculation agrees well with one done 16 years later in 2008 [50]. Liu et al. also looked at diffusion on the (311) and (331) planes, where movement was one-dimensional. On Cu(311) they found a barrier of 0.26 eV with a prefactor of 1.2 × 10−3 cm2/sec (AFW) and 0.28 eV and a prefactor of 3.1 × 10−3 cm2/sec (VC). On Cu(331), they estimated a barrier of 0.28 eV and a prefactor of 1.0 × 10−3 cm2/sec using AFW potentials; with VC interactions these figures were 0.33 eV and 3.0 × 10−3 cm2/sec. Using EMT, Hansen et al. [39] offered calculations for motion along the channels on Cu(110), with a barrier of roughly 0.18 eV, and across channels by atom exchange requiring an energy around 0.27 eV. Perkins and DePristo [28] used the CEM method in their work. They investigated the influence of the number of active atoms in the calculations of the energy. For in-channel diffusion on Cu(110) with 113 active atoms they found a barrier of 0.260 eV, and 0.485 eV for cross-channel motion by exchange of atoms using MD/MC-CEM. When the number of active atoms decreased to 13 the energy increased to 0.294 eV for hopping and 0.489 eV for exchange. From CEM they got much lower values of 0.08 eV for hopping and 0.09 eV for exchange in a system with 59 active atoms, which are rather too low. Calculations were also made by Stoltze [27] with effective medium theory. He found a barrier to in-channel diffusion of 0.292 eV and 0.419 eV for atom exchange to give cross-channel motion. Cross-channel atom jumping required overcoming a barrier of 0.826 eV. Karimi et al. [40] again relied on EAM interactions with AFW parameters and found a barrier to in-channel motion of 0.24 eV; for exchange with atom transfer to the adjacent channel they obtained 0.30 eV, but what was surprising in this study is that for exchange with atom transfer to the same channel the barrier was much higher, 0.87 eV. That may explain the preferential direction for exchange obtained from the displacement distribution on Ir(110) [41]; however, the physics of this process is not completely understood. Jumping across channels is definitely an unfavorable event with a barrier of ~1.15 eV. In 1997 Merikoski et al. [36] looked at atomic processes next to steps using molecular dynamics based on effective medium theory. For hopping on Cu(311) they found a barrier of 0.232 eV. Hopping across to an adjacent channel required 0.889 eV while exchange needed only 0.822 eV. Once more hopping was favorable on fcc(311) surfaces. Evangelakis et al. [42] examined the movement of Au on Cu(110) in the same year. For comparison they also determined the barriers for self-diffusion: 0.47 eV for hopping and 0.63 eV for exchange. One year later, the same authors derived much smaller values [43], again using RGL interactions, a second-moment approximation of the tight-binding model, to do molecular dynamics simulation of self-diffusion; their results are shown in Fig. 4.11. For temperatures up to 750 K they observed jumps along the channels; from the slope of their Arrhenius plot they derived a barrier of 0.25 eV; cross-channel motion via atom exchange took place over a barrier of 0.30 eV. Hopping is a slightly more favored mechanism in this range of temperatures. Above 850 K the frequency of both mechanisms has saturated and diffusion involves more complicated and concerted movements of hopping and exchange. No information about frequency prefactors was derived. RGL interactions were utilized in calculations by Mottet et al. [44] for self-diffusion on Cu(110). They found a barrier of 0.23 eV for in-channel motion and 0.29 for crosschannel movement by atom exchange. The prefactor for the diffusivity was not

196

Diffusion on one-dimensional surfaces

1

Frequency ν (THz)

Hopping ED = 0.25 eV 0.1

0.01 Exchange ED = 0.30 eV 0.001 10

Fig. 4.11

15

20 1/kT (eV–1)

25

30

Arrhenius plots for hopping and cross-channel exchange in self-diffusion on Cu(110), based on molecular dynamics simulations using RGL potentials (after Evangelakis et al. [43]). –2

ED2 = 0.16 ± 0.01 eV ν2 = 2 × 1012 sec–1

–3

In R

–4

–5

–6

–7 1.5

Fig. 4.12

ED1 = 0.213 ± 0.007 eV v1 = 9 × 1012 sec–1 3 2 2.5 1000/T (K–1)

3.5

Dependence of the diffusivity of Cu on Cu(110) upon the reciprocal temperature. Results obtained by molecular dynamics simulation with RGL potentials. Low temperature results (solid line) and high temperature (dash-dotted) are plotted separately, with activation energies of 0.213 ± 0.007 eV and 0.16 ± 0.01 eV respectively (after Montalenti and Ferrando [45]).

calculated. This study was continued by Montalenti and Ferrando [45] and the same values were reported. However, a detailed plot of the jump rates for temperatures from 300 K to 600 K revealed that a single straight line Arrhenius plot did not do justice to the data, which divided into two parts, below and above 400 K, as shown in Fig. 4.12. The low temperature results yielded an activation energy of 0.213 ± 0.007 eV, with a frequency prefactor of 9 × 1012 sec−1; in the higher temperature range, the barrier was smaller, 0.16 ± 0.01 eV with a prefactor of 2 × 1012 sec−1. At the highest temperature of 600 K, double jumps amounted to 14% of the total. It is interesting to note that double

4.4 Copper: Cu(110), Cu(311), Cu(331)

197

jumps caused a decrease of the diffusivity not an increase! It is also worth noting the standard frequency prefactor for both temperature ranges. Prévot et al. [46] also resorted to RGL calculations of the diffusion barriers on Cu(110). For in-channel motion of Cu atoms they found an activation energy of 0.251 eV, whereas Cu-Cu exchange involved a barrier of 0.284 eV. The same year Wang and Adams [47] worked on kinetic Monte Carlo modeling of growth on copper surfaces. For estimates of diffusion barriers they used the EAM developed by Daw and Baskes. On the Cu(110) surface they obtained a value of 0.23 eV for jumps inside the channel and 0.30 eV for perpendicular movement by exchange. Again applying RGL potentials to the problem of self-diffusion by hopping on the (110) plane of copper, Kürpick [31] found an activation energy of 0.25 eV with a prefactor of (1.1 ± 0.1) × 10−3 cm2/sec. Agrawal et al. [7] have made similar estimates, but with Lennard-Jones potentials that give only two-body interactions. For in-channel motion they found an activation energy of 0.76 eVand a prefactor of 7.5 × 10−3 cm2/sec; for jumps between neighboring channels the barrier was significantly higher, 1.31 eV, with a prefactor of 2.6 × 10−2 cm2/sec. As usual for studies with only pair interactions, the in-channel value for the diffusion barrier is too high. Ndongmouo and Hontinfinde [32] again used RGL interactions in their simulations and for in-channel motion attained a barrier of 0.23 eV. For cross-channel atom exchange the barrier was 0.27 eV, and for jumps between neighboring channels 1.07 eV. In molecular dynamics simulations also relying on RGL potentials, Kim et al. [34] found a barrier of 0.241 eV for in-channel motion, 0.323 eV for cross-channel exchange, and 1.020 eV for cross-channel jumps. Calculations with EAM potentials were done by Yildirim et al. [48], who found the barrier to in-channel motion was 0.23 eV; the prefactor at 300 K amounted to 6.29 × 10−4 cm2/sec. Jumping across a channel required 1.146 eV, with a prefactor of 9.97 × 10−4 cm2/sec; exchange was not considered in this investigation. The same energetics were reported again by Durukanoglu et al. [49] for in-channel and cross-channel hopping. Atom exchange required an activation energy of 0.34 eV, so that in-channel hopping was the preferred process. The prefactor for hopping did not change much with temperature, becoming 6.39 × 10−4 cm2/sec at 600 K. This work has been challenged, however, by Kong and Lewis [33,50], who used the same EAM potentials as Yildrim et al. but evaluated the total density of states. For the barrier height along the channels they found a value of 0.23 eV, for cross-channel jumps the value was 1.146 eV, both values are in exact agreement with the results of Yildrim et al. For the prefactor for an atom diffusing at 300 K they calculated a value of 4.0 × 10−3 cm2/sec by summing over all vibrational frequencies compared to 1.4 × 10−3 cm2/sec if only local vibrations are taken into account. The same values were measured at 600 K. Kong and Lewis concluded that the local approximation was not always appropriate in calculating the prefactor. Recently, Stepanyuk et al. [51] investigated growth of cobalt nanostructures on a Cu(110) surface. For Cu in-channel movement they derived a value of 0.26 eV from molecular statics, exchange required only slightly more − 0.3 eV. Exchange of Cu close to an embedded Co atom required 0.1 eV less. It is regrettable that no reliable experimental work has been done to determine the selfdiffusion characteristics of Cu(110). From the work of Robinson et al. the exchange

198

Diffusion on one-dimensional surfaces

mechanism is likely to be present on the surface. The calculated values for the selfdiffusion coefficients are in reasonable shape – the results of the various efforts using reasonable interactions between the atoms are close together. It is clear that many-body interactions are necessary to consider diffusion. Based on the theoretical studies, the exchange mechanism seems to be a bit more energetically demanding for diffusion, but the difference in barriers is not big. A combination of both may be possible but will need direct experimental proof, but the work of Robinson et al. seems to point in this direction.

4.4.2

Hetero-diffusion Several investigations have been done of silver diffusion on copper surfaces. However, the concentrations of silver in these studies were quite appreciable and it is not clear that information about single atom phenomena was determined and that interactions were insignificant. Among these reports is the work of Roulet [52], Ghaleb and Peraillon [53], and Kürpick et al. [54]. What is worth noting, however, is the suggestion [53] of an exchange movement for Ag on Cu(110), based on observations of the anisotropy of diffusion independent of temperature. Work has been reported for diffusion of lead on Cu(110) by Prévot et al. [55] using Rutherford backscattering spectrometry. In their studies a square of the surface was covered with ~ 0.1 monolayer of deposited lead, and the spreading of this deposit, at temperatures from 500 K to 800 K was then measured, as shown in Fig. 4.13. The diffusion coefficients along and across the channels at the different temperatures is given in Fig. 4.13b, yielding for motion along the channels the result D ¼ 1:5  10
2 expð
0:57 eV=kTÞcm2 = sec :

(4:1)

For cross-channel movement the diffusivity was D ¼ 6:2  10
3 expð
0:57 eV=kTÞ cm2 = sec :

0.12

Coverage Θ (ML)

0.10 0.08 0.06

Diffusion profile along [001]

Diffusion profile along [110]

0.04 0.02 0 –6

Fig. 4.13

(a) –4

–2 0 2 Position (mm)

4

6

10–5

–2

2

–3

2

Dx [110]; ED = 0.57 eV; Do = 1.5 × 10 cm /sec

Diffusivity D(cm2/sec)

Initial concentration profile

(4:2)

Dy [001]; ED = 0.57 eV; Do = 6.2 × 10 cm /sec

10–6

10–7 (b) 10–8 1.2

1.4

1.6 1000/T (K–1)

1.8

2.0

Diffusion of Pb on Cu(110) plane measured by Rutherford backscattering. (a) Lead concentration profiles on Cu(110) plane. In-channel motion is more rapid than cross-channel diffusion. (b) Arrhenius plots for Pb diffusivities on Cu(110). Activation energies are the same in the two directions, but prefactor is larger for in-channel motion (after Prévot et al. [55]).

199

4.4 Copper: Cu(110), Cu(311), Cu(331)

The activation energies are the same, but the prefactor along the channel is higher than across. To rationalize this behavior, Prévot et al. came up with a model in which at temperatures above 250 K the lead atoms are embedded in the outermost layer; for lead to diffuse, it must first emerge from being embedded and change to an adatom by atom exchange. This exchange occurs with a copper atom, and not with another lead, as the diffusivity is not dependent on the coverage of lead atoms. The lead adatoms then diffuse along the channels until they again embed. This model was soon tested by Prévot et al. [46], who carried out static as well as molecular dynamics studies using RGL inter-atomic potentials. From static energy calculations they found an activation energy of 0.212 eV for lead diffusion along the channels; for reinsertion of the lead by exchange with a copper lattice atom the barrier was 0.281 eV. The activation energy for emergence of a lead atom by exchange with a copper adatom was obtained as 0.590 eV, quite close to the experimental value of 0.57 eV. For diffusion of copper atoms along the channels the barrier was 0.251 eV, and for exchange with copper 0.284 eV. Molecular dynamics simulations for lead atoms at temperatures between 400 K and 700 K showed that multiple jumps appeared frequently and it was possible to obtain the rate of single jumps  α ¼ 4:9  1011 expð
0:14  0:02 eV=kTÞ sec
1 ;

(4:3)

the rate of correlated jumps  ce ¼ 8:9  1011 expð
0:18  0:02 eV=kTÞ sec
1 ;

(4:4)

and finally the rate for returning lead to the outer layer  r ¼ 3:7  1012 expð
0:27  0:03 eV=kTÞ sec
1 :

(4:5)

These results are shown in the Arrhenius plots in Fig. 4.14a. The value of the barrier for reinsertion from the Arrhenius plot is close to the static one: 0.27 eV compared to Temperature T(K) 700

600

500

400 1 (b) 0.8 Ratio

Frequency ν(Hz)

1011

(a) 1010

109 1.4

να νr νce νall

0.6

0.4 1.6

1.8

2.0

2.2

1000/T (K–1) Fig. 4.14

2.4

2.6

400

600

500

700

T(K)

(a) Dependence of jump rates for Pb on Cu(110) upon temperature, obtained with RGL potentials. Shown are single jumps, reinsertions, correlated events, all events. (b) Ratio of multiple (open circles) and of correlated events (full squares) to single jumps for in-channel diffusion (after Prévot et al. [46]).

200

Diffusion on one-dimensional surfaces

Frequency ν (1012sec–1)

1200

Temperature T(K) 800

600

Exchange ED = 0.23±0.04 eV 10–1 Hopping ED = 0.19±0.03 eV

10–2 10

Fig. 4.15

1000

12

14 16 1/kT (eV–1)

18

20

Arrhenius plot for the jump frequency of Au atoms on Cu(110) by hopping along the channels and by atom exchange. Data derived by molecular dynamics simulations with RGL potentials (after Evangelakis et al. [42]).

0.281 eV. But the barrier for single jumps (0.14 eV compared to 0.212 eV for the static value) is quite different. The authors attribute this difference to not taking into account multiple jumps and other events. They also made molecular dynamics estimates and found long transitions in addition to single jumps. As shown in Fig. 4.14b, increasing the temperature increased the number of multiple jumps, but it should be noted that the temperatures were quite high. Molecular dynamics simulations of the diffusion of gold atoms on Cu(110) were done using RGL interactions by Evangelakis et al. [42]. They observed both atom hopping along the channels and exchange with lattice atoms. The frequency of the diffusion processes is plotted in Fig. 4.15 against 1/kT, yielding a diffusion barrier of 0.19 ± 0.03 eV for hopping and 0.23 ± 0.04 eV for exchange. It must be noted, however, that in the temperature range from 700 K to ~1100 K, exchange occurs more frequently than hopping. In a similar way, Kim et al. [34] arrived at a barrier height of 0.244 eV for in-channel diffusion of Ni on Cu(110), 0.318 eV for Pd, 0.254 eV for Ag, 0.376 eV for Pt, and 0.333 eV for Au. Exchange was not taken into account. The last value for gold is quite different from that derived by Evangelakis et al. [42], despite the similarity of the methods, but otherwise there are not a sufficient number of measurements available to allow a comparison. Recently Stepanyuk et al. [51] looked at the movement of cobalt on Cu(110) with molecular statics and with VASP. From molecular statics they derived a value of 0.29 eV for in-channel jumps, while exchange required overcoming a barrier of 0.30 eV; perpendicular jumps were unlikely with an energy higher than 1 eV. From VASP the diffusion barriers were slightly higher, 0.35 eV for in-channel motion and 0.32 for exchange.

4.5

Molybdenum: Mo(211) Unfortunately there is no direct experimental information available about single atom movement on the molybdenum (211) surface. The only information concerns spreading material over the surface and monitoring the work function changes. A lithium circular

201

4.5 Molybdenum: Mo(211)

0.8

3

p(1×1)

(a)

Iog Do (cm2/sec)

0.6 ED (eV)

(b)

2

0.4 p(1×4) p(1×2) 1D-IS

0.2

1 0 –1

0.0 0.0

1.5

–2 0.0

1.0

1.5

Coverage Θ

Dependence of the diffusion characteristics for Li in-channel diffusion on Mo(112) upon coverage. (a) Activation energy for diffusion. (b) Prefactor Do (after Naumovets et al. [56]).

IogDo

1.2

0.6 0.0

1.0 0.0

–2.0

0.1

0.2

PT-I 0.3 0.4 Coverage Θ

(4 × 2)

0.8

2.0

–1.0

ED p(1 × 5)

1.0

p(1 × 9)

ED (eV)

T > 660 K T < 660 K

Fig. 4.17

0.5

0.5

Iog Do (cm2/sec)

Fig. 4.16

0.5 1.0 Coverage Θ

C-I 0.6

Activation energy and prefactor for diffusion of Sr on Mo(211) channels as a function of coverage (after Loburets et al. [57]).

spot was deposited on the surface by Naumovets et al. [56] and a strong anisotropy of spreading was observed. The results were analyzed using the Boltzmann–Matano method. Coverages as low as 0.02 ML were investigated, which might yield information about single atom movement. For low coverage, the activation energy for motion along channels was around 0.6 eV, and was coverage independent up to 0.5 ML, as shown in Fig. 4.16a. In Fig. 4.16b the dependence of the prefactor for the diffusivity on the coverage is presented; at low concentrations the prefactor Do amounted to 2.5 cm2/sec. At the same time, Loburets et al. [57] studied diffusion of Sr on Mo(112) using the same technique. The activation energy and prefactor for the diffusivity are shown as a function of coverage in Fig. 4.17, with the former at 0.9 eV at the lowest concentration. The same method was also used by Loburets et al. [58] to investige diffusion of a submonolayer of Dy on Mo (211). However, for Dy the activation energy increased in going to low coverages, reaching 1.4 eV, as shown in Fig. 4.18. This method does not account for controlling defects on the surface, so it is impossible to get information about the mechanism of surface diffusion. In 1999, Loburets [59] studied the diffusion of copper on the (211) surface of molybdenum, again measuring changes in the electronic work function of the substrate. Work appears to have been done at reasonably low concentrations, < 0.1 ML, so that

202

Diffusion on one-dimensional surfaces

4 (a)

ED(eV)

–7.5 Θ = 0.35

–8.0

(b)

Θ = 0.1 Dy/Mo(211)

–2

ED 0.5

–9.0 1.25 Fig. 4.18

0

1.5 1.0

Θ = 0.10

–8.5

2

Do

log Do(cm2/sec)

log D (cm2/sec)

–7.0

1.30 1.35 1000/T (K–1)

1.40

1.45

0.0

0.1

0.2 0.3 0.4 Coverage Θ

0.5

0.6

Diffusion of Dy on Mo(112) plane. (a) Arrhenius plots for the diffusivity at different coverages. (b) Diffusion characteristics in their dependence on coverage (after Loburets et al. [58]).

individual adatom behavior may be dominant in the activation energy of 0.45 eV and the prefactor Do = 1.2 × 10−3 cm2/sec. Unfortunately this plane was not investigated in FIM or STM studies, and there are also no theoretical investigations available for a comparison.

4.6

Rhodium: Rh(110), (311), (331) Measurements of diffusion on rhodium by Ayrault [60] in 1974 came not too long after the initial work on tungsten surfaces, and should therefore be treated carefully. However, measurements were done with a single atom deposited on each plane. Arrhenius plots for these systems are shown in Fig. 4.19. On the (110) plane, diffusion of rhodium occurred over a barrier of 0.60 ± 0.03 eV and a prefactor of 3 × 10−1 cm2/sec. Diffusion over (311) involved a smaller activation energy, 0.54 ± 0.04 eVand a prefactor Do = 2 × 10−3 cm2/sec. On the (331) plane, the barrier to diffusion was 0.64 ± 0.04 eV, higher than for the other two planes, and the prefactor amounted to 1 × 10−2 cm2/sec. What is interesting is that on all these surfaces, diffusion was always in the direction of the channels, and occurred by simple jumps. Unfortunately no other measurements were done later on for self-diffusion of rhodium, so these data stand alone. Almost twenty years later, Kellogg [61] also with the FIM examined the movement of a platinum atom on these rhodium planes. Only on Rh(311) was an Arrhenius plot given, Fig. 4.20, with an activation energy of 0.44 ± 0.05 eV and a prefactor Do = 1.6 × 10−4 ± 0.05 cm2/sec. Movement on this plane proceeded by simple hopping along the channels. The activation energy, but also the prefactor, are significantly lower than for the self-diffusion of rhodium. On (331) the activation energy, based on four temperatures from 256 K to 278 K, was given as 0.72 ± 0.02 eV, presuming a prefactor of 10−3 cm2/sec. On (110), the activation energy for cross-channel transitions was estimated as ~ 0.65 eV from the sudden loss of Pt atoms, which shows that the movement proceeded by exchange not by hopping. Cross-channel movement was certain, but the extent of inchannel motion was not clear. This study showed that the exchange mechanism is not forbidden for rhodium surfaces even if self-diffusion proceeds by hopping.

203

4.6 Rhodium: Rh(110), (311), (331)

7

320 300

220

210

Temperature T(K) 180 200 190

60

55

50

6 (110)

(111)

In (N 02)

5 4 3

(311) (100) (331)

2 1 0 3.0 Fig. 4.19

3.5

4.5

5.0 5.5 1000/T (K–1)

16

17

18

19

20

Self-diffusion of rhodium adatoms on Rh planes determined in FIM [60]. On (111) and (100) planes motion is two-dimensional.

In [<x 2>/2t) (cm2/sec)]

–32 Pt on Rh(311) –34 –36 –38 –40 –42 4.8 Fig. 4.20

ED = 0.44 ± 0.05 eV Do = 1.6 × 10–4±0.5 cm2/sec 5.0

5.2

5.4 5.6 1000/T (K–1)

5.8

6.0

6.2

Dependence of Pt atom diffusivity on Rh(311) plane upon the reciprocal temperature. Results from FIM observations of atom displacements (after Kellogg [61]).

In 1980 Flahive and Graham [20] used Morse potentials to calculate diffusion energies shown in Fig. 4.9. Approximation with Morse potentials gives a result a bit lower than the experimental value. Calculations for self-diffusion on Rh(110) were done by Liu et al. [2], who found a barrier of 1.24 eV for hops along the channel with a prefactor of 8 × 10−3 cm2/sec using Lennard-Jones potentials. With the same potentials they calculated a barrier of 0.64 eV on the (311) with a prefactor of 5.4 × 10−3 cm2/sec, and on the (331) plane the results were 1.31 eV and 7.4 × 10−3 cm2/sec. The first and last barriers appear rather high. The barriers calculated with Morse potentials were much smaller and amounted to 0.48 eV, 0.44 eV, and 0.62 eVon the (110), (311), and (331), and are in better accord with experiments. Unfortunately, Liu et al. did not use a many-body EAM potential

204

Diffusion on one-dimensional surfaces

for rhodium. Agrawal et al. [7], who also utilized Lennard-Jones interactions, came up with an activation energy of 1.20 eV and a prefactor Do = 4.4 × 10−3 cm2/sec for self-diffusion along the channels on Rh(110), but this is clearly much too high a value. Their estimates for cross-channel jumps were 2.06 eV for the barrier and 1.0 × 10−2 cm2/sec for the prefactor. It appears that the Lennard-Jones potential is unsuitable for diffusion on Rh surfaces. The first and so far only calculations for self-diffusion of rhodium on Rh(110) with RGL many-body potentials was done by Ndongmouo and Hontinfinde [32]. They calculate an activation energy of 0.78 eV for jumps along the channels, rather higher than experiments, and 0.91 for cross-channel motion by atom exchange. Jumps across channel required going over a barrier of 1.80 eV. Theoretical results for channeled rhodium planes are not in good shape. The experiments, while not really definitive, appear reasonable. The calculations with Morse potentials or from RGL are in reasonable accord, but the other estimates are quite far from the experimental results and more work here is clearly desirable.

4.7

Palladium: Pd(110), (311), (331) Although experimental self-diffusion studies have not been made on Pd(110), the growth of copper has been examined by Bucher et al. [62], who studied the saturation density of copper islands at a coverage lower than 0.1 ML over a range of temperatures. Their results, in Fig. 4.21, show two different processes. At temperatures below 350 K, Temperature T(K) 420 350 300 250 220 200 180

104

Island Density

ED[001] = 0.75 eV 10

3

102

ED[110] = 0.51 eV

101

100

3

5

4

6

–1

1000/T (K ) Fig. 4.21

Copper island density on Pd(110) plotted against reciprocal temperature for both transverse density (upper curve) and overall density, giving activation energies for diffusion (after Bucher et al. [62]).

4.7 Palladium: Pd(110), (311), (331)

Fig. 4.22

205

Comparison of STM images of Cu islands on Pd(110) with simulations at (a) 265 K, (b) 300 K, and (c) 320 K. In-channel barrier height was 0.3 eV, cross-channel barrier 0.45 eV (after Li et al. [65]).

one-dimensional islands form along the channels, and Bucher et al. derived an activation energy for in-channel diffusion of 0.51 ± 0.05 eV. Above 300 K, the island density across the channel direction, shown by the upper curve in Fig. 4.21, yielded an Arrhenius plot with a barrier of 0.75 ± 0.07 eV. However, the values may not represent single atom movement, due to possible long-range interactions at the investigated coverage. Bucher et al. also did calculations with EAM interactions [62–64] to find barriers for diffusion of Cu by hopping along and across the channels of 0.32 eVand 1.25 eV; the latter value was recognized as not realistic. Instead the authors suggested a series of steps including Cu-Pd exchange with an overall barrier height of 0.57 eV. What is interesting is that exchange of a Pd adatom with a Cu atom embedded in a row of palladium atoms costs only 0.42 eV; Pd-Pd exchange required the same energy as Cu-Pd exchange, 0.57 eV. Hopping of a Pd adatom which emerged after exchange cost 0.33 eV. Massobrio and Fernandez [63] examined diffusion of palladium atoms on Pd(110) relying on EAM potentials and calculated an activation energy of 0.28 eV for in-channel hopping, 0.42 eV for cross-channel exchange, and a very high barrier of 1.56 eV for cross-channel jumps. In 1997 Li et al. [65] developed a kinetic model to describe the temperature dependence of island shapes observed in the STM. For islands of Cu on Pd(110) comparisons of experiment and model predictions are shown in Fig. 4.22. From the island densities plotted in Fig. 4.23 they derived activation barriers of 0.3 eV for diffusion along the channel and 0.45 eV for movement across channels. They also established that corner rounding was very important in the investigation of island shape; corner rounding slowed the overall process due to its high activation energy of 0.65 eV. In 1980 Flahive and Graham [20] used Morse potentials to calculate the activation energy for self-diffusion on palladium. Their results are summarized in Fig. 4.9. Taking into account only pair interactions, they arrived at values a bit higher than later estimates: for Pd(110) around 0.45 eV, Pd(311) ~0.35 eV, and Pd(331) roughly 0.55 eV. Calculations of self-diffusion on Pd(110) were made with EAM many-body potentials by Liu et al. [2]. For in-channel motion the activation energy was 0.28 eVand the prefactor 4 × 10−4 cm2/sec using AFW potentials. With VC potentials they found a diffusion barrier of 0.30 eV and a

206

Diffusion on one-dimensional surfaces

Island Densities

104 103 102 101 100

2

3

4

5

6

1000/T (K–1) Fig. 4.23

Copper island densities on Pd(110) surface in their dependence upon the reciprocal temperature. Linear cross-channel islands indicated by circles, total island density by squares; experimental results plotted with open circles, solid symbols give simulated findings (after Li et al. [65]).

prefactor of 3.5 × 10−3 cm2/sec. For cross-channel motion by atom exchange, the activation energy was 0.42 eV with a prefactor 3.3 × 10−2 cm2/sec (AFW), and 0.34 eV with a prefactor of 2.4 × 10−2 cm2/sec (VC). Liu et al. also evaluated diffusivities on the (311) and (331) planes. On the first, the activation energy was 0.37 eV and the prefactor 1.2 × 10−3 cm2/sec (AFW), and 0.41 eV with a prefactor of 3.1 × 10−3 cm2/sec (VC). On the (331) plane the barrier was 0.33 eV with 8 × 10−4 cm2/sec (AFW) and 0.37 eV and a prefactor of 2 × 10−3 cm2/sec (VC). Using CEM and MD/MC-CEM methods, Perkins and DePristo [28] calculated the barrier to in-channel as well as to cross-channel movement, by exchange, on Pd(110). For in-channel diffusion the barrier calculated with CEM was 0.30 eV, and 0.33 for atom exchange; using MD/MC-CEM they found a similar activation energy of 0.28 eV for in-channel motion and 0.38 eV for cross-channel movement with 113 atoms active. The barrier was around 0.04 eV higher using only 13 atoms active on the surface: 0.32 eV for hopping and 0.42 eV for exchange. Prefactors for the diffusivity were not investigated. Stoltze [27], with effective medium theory, arrived at a barrier for self-diffusion along the channels of 0.366 eV, and 0.599 eV for cross-channel exchange; cross-channel jumps required a higher energy of 0.776 eV. Depending on Lennard-Jones potentials, Agrawal et al. [7] found a barrier to Pd diffusion along the channels of Pd(110) of 0.72 eV with a prefactor Do = 1.8 × 10−3 cm2/sec; for cross-channel jumps the activation energy was 1.35 eV, with a prefactor of 9.2 × 10−3 cm2/sec. These estimates are of course unrealistic. Ndongmouo and Hontinfinde [32] used RGL potentials in their calculations for Pd atoms on Pd(110) to arrive at an in-channel barrier of 0.39 eV and 0.54 eV for crosschannel motion by atom exchange. For jumps between adjacent channels the activation energy was higher, 0.99 eV. No studies of prefactors were performed. Similar barriers have been obtained recently by Kim et al. [34] using molecular dynamics simulations and RGL interactions. For in-channel self-diffusion they found a barrier of 0.380 eV, 0.551 eV for exchange, and 0.965 eV for cross-channel jumps. In-channel diffusion of

207

4.8 Silver: Ag(110), (311), (331)

Ni involved a barrier of 0.369 eV, 0.315 eV for Cu, 0.333 eV for Ag, 0.520 eV for Pt, and 0.392 eV for Au. Again no information about prefactors is available in this study, nor about the possibility of the exchange mechanism for hetero-diffusion. The theoretical estimates available for self-diffusion on Pd(110) are quite close together, but regrettably there is no information available from experiments. Values for the movement of Cu on Pd(110) surfaces obtained from MC analysis of STM data by Li et al. agree nicely with results from EAM by Bucher et al. [62] and RGL by Kim et. al. [34].

4.8

Silver: Ag(110), (311), (331) The mobility of silver atoms on the (110) plane of silver has been examined in a variety of ways. Morgenstern et al. [66] looked at the decay of Ag islands on Ag(110) in STM studies, together with KMC simulations relying on molecular dynamics estimates of the diffusion energetics with RGL potentials. From these simulations they found values of 0.279 eV for the barrier to in-channel diffusion and 0.394 for cross-channel movement. Corner rounding was also considered in these investigations. In 2001, De Giorgi et al. [67] did scanning tunneling microscope studies of homoepitaxial growth on Ag(110). The saturation island density, examined for a deposit of 0.16 monolayers (ML) at temperatures from 140 K to 200 K, is shown in Fig. 4.24. Together with KMC simulations developed earlier by Mottet et al. [44], shown in Fig. 4.25, they came up with an in-channel barrier to self-diffusion of 0.30 ± 0.03 eV. A cross-channel barrier was estimated from island rotation at low temperature to be at least 0.05 eV higher than for in-channel diffusion. Frequency prefactors were fixed at a value of 1013 sec−1. However, the results may not represent single atom movement due to the high coverage and interactions between atoms. This work was followed by a study from Pedemonte et al. [68] in 2002, who did quasielastic scattering experiments using a helium atom beam having a most probable energy of 0.0073 eV. From the width of the quasi-elastic peak shape at surface temperatures from 600 to 750 K, they were able to find the diffusivity at the different temperatures, as shown in Fig. 4.26. A barrier to diffusion of 0.19 ± 0.05 eV was found for movement along the

Island Density

10–2

10–3 STM data simultation data

10–4

4

Fig. 4.24

5

6 1000/T (K–1)

7

8

Ag island density on Ag(110) plane as a function of reciprocal temperature. From KMC simulations with RGL potentials, in-channel diffusion barrier was 0.30 ± 0.03 eV (after De Giorgi et al. [67]).

208

Diffusion on one-dimensional surfaces

Fig. 4.25

Comparison of STM images of Ag islands on Ag(110) with KMC simulations at (a) 140 K (50 × 50 nm2), (b) 170 K (100 × 100 nm2), and (c) 210 K (100 × 100 nm2), showing good agreement between the two (after De Giorgi et al. [67]).

Temperature T(K) 70 60

750

650

700

600

ED~0.19 eV

50

Γε (μeV)

40 30 ED = 0.19±0.05 eV 20 QHAS data MD simulation 10 1.3

Fig. 4.26

1.4

1.6 1.5 1000/T (K–1)

1.7

Arrhenius plot of the diffusivity of Ag atom on Ag(110), determined by quasi-elastic He atom scattering. Simulations from Ferrando [76] (after Pedemonte et al. [68]).

209

4.8 Silver: Ag(110), (311), (331)

Diffusivity D(10–5 cm2/sec)

T (K) 775 750

700

650

2.0 1.5

1.0

ED = 0.29±0.02 eV 1.30

1.40

1.50

1/T (10–3 K–1) Fig. 4.27

Dependence of Ag diffusivity on Ag(110) plane upon reciprocal temperature, derived from energy resolved scattering of He atoms. Barrier for in-channel motion 0.29 ± 0.02 eV (after Pedemonte et al. [69]).

channels, that is in the direction. Movement of atoms in this range of temperatures was by single jumps only. This technique does not provide the possibility of controlling the number of adatoms on the surface, nor their distance or information about defects, so the work has to be treated carefully. Further studies were published by Pedemonte et al. [69,70], who continued their helium scattering work. The experiments on Ag(110) were extended to temperatures from 650 K to 775 K, and they arrived at a higher self-diffusion barrier of 0.26 ± 0.03 eV along the channels. Motion across the channels, in the direction, was almost undetectable even at 775 K. Pedemonte et al. still claimed that movement proceeded by single jumps along the channels on the surface, but did not provide an explanation of what gave a higher activation energy at higher temperatures. Additional scattering studies were done by Pedemonte et al. [71], with the upper limit at 800 K, to arrive at the Arrhenius plot of silver diffusion along the channels in Fig. 4.27; the barrier proved to be even higher than in the previous study of the same group, 0.29 ± 0.02 eV. Although diffusion by cross-channel transitions was essentially undetected up to 750 K, such transitions were identified at 800 K and it was concluded that the atom was moving by exchange. The energetics of atom exchange were not determined. One possible reason for these observations could be a change in the coverage of adatoms at the surface with increasing temperature, as the concentration was never specified in this study. If this is the case the activation energy in the lowest temperature range should be closest to single atom movement at the surface, but other factors might play a role as well. Quite interesting data were reported for gold atoms deposited on the Ag(110) surface. Not only was exchange proposed by Johnson [72], but the burying of gold atoms with hole creation was observed by STM [73,74]. These effects were not observed for silver atoms on this surface. A significant number of attempts have been made to calculate the diffusion characteristics on silver. The first were by Flahive and Graham [20] using Morse potentials, summarized in Fig. 4.9. Liu et al. [2] used many-body EAM potentials and estimated an activation energy for self-diffusion along the channels on Ag(110) of 0.32 eV with a

210

Diffusion on one-dimensional surfaces

prefactor of 1.0 × 10−3 cm2/sec, using AFW parameters. In the VC approximation the barrier proved to be 0.25 eV and the prefactor 2.7 × 10−3 cm2/sec. For cross-channel motion the barriers were higher, 0.42 eV, and the prefactor 4.0 × 10−2 cm2/sec (AFW) and 0.31 eV and 2.5 × 10−2 cm2/sec (VC). Calculations were also made for other planes. On the (311) surface the diffusion barrier was 0.26 eV and the prefactor 1.0 × 10−3 cm2/ sec (AFW parameters), and again 0.26 eV and 3.0 × 10−3 cm2/sec (VC parameters). For the (331) plane the barriers were derived as 0.34 eV with a prefactor of 1.0 × 10−3 cm2/sec (AFW) and 0.29 eV and a prefactor of 2.0 × 10−3 cm2/sec (VC). Perkins and DePristo [28] also made diffusion estimates on Ag(110). With their MD/ MC-CEM method they reported an in-channel diffusion barrier of 0.248 eV, and 0.328 eV for cross-channel exchange. Findings from CEM did not differ much; they obtained 0.26 eV and 0.34 eV, in reasonable agreement with the work above. They also looked at the influence of the number of active atoms taken into account in the simulations; when this number was only 13, the activation energies were higher; for in-channel hopping the barrier was 0.29 eV and for exchange 0.371 eV. In the same year, Stoltze [27] in simulations with EMT potentials found an activation energy of 0.291 eV for jumps along the channels, 0.561 eV for atom exchange, and 0.639 eV for jumps across the channel. Hontinfinde et al. [75] used RGL interactions to again estimate the self-diffusion barriers on Ag(110). For in-channel single jumps they found an activation energy of 0.28 eV, in very good agreement with the results of Pedemonte et al. [71]. The barrier for cross-channel exchange was 0.38 eV. Cross-channel hops required a much higher energy of 0.82 eV. Ferrando [76] extended this study to investigate diffusion mechanisms. He claimed that at any temperature single transitions represented 90% of the jumps, and that the longer jumps were doubles. Around 600 K more complicated mechanisms started to play a role as well, and a sudden decrease in the number of single jumps was observed at higher temperatures. The same values were again given by Montalenti and Ferrando [45], who also considered the presence of long jumps; at 600 K, 8.4% of the transitions were double jumps, but at 450 K the percentage diminished to 2.8. In 1999 Rusponi et al. [77] looked at vacancy and adatom islands created by ion sputtering of Ag(110) over a range of temperatures to explain the shape evolution of the islands. They used KMC simulations, but for the starting values of the energy did quenched molecular dynamics simulations based on RGL potentials. They used the same diffusion barriers on Ag(110) as previous investigators, 0.28 eV and 0.38 eV for in-channel and cross-channel diffusion. Calculations of the activation energy of self-diffusion were made by Agrawal et al. [7] using Lennard-Jones potentials. They found a high barrier of 0.59 eV and a prefactor Do = 2.6 × 10−3 cm2/sec for in-channel motion, and 1.10 eV with a prefactor of 6.9 × 10−3 cm2/sec for cross-channel events, quite far away from previous investigations. Merikoski et al. [36] studied self-diffusion on the Ag(311) surface using EMT potentials. For hopping along the channels, they found a barrier of 0.220 eV; crosschannel exchange required overcoming a barrier of 0.727 eV, whereas cross-channel jumps turned out to have a comparable energy to exchange − 0.704 eV. Nie et al. [78] have done density-functional calculations for the self-diffusion of silver atoms on Ag(110)-(2 × 2). On the bare surface in-channel diffusion of Ag occurred over a barrier of 0.266 eV. For cross-channel jumps the barrier was much higher, amounting to

4.9 Tungsten: W(211)

211

0.757 eV, but exchange required only ~0.4 eV, so that in-channel motion was clearly preferred. Vacancy formation cost about 0.275 eV. Ag diffusion on an Sb covered Ag(110) surface required 0.275 eV for in-channel movement and 0.673 eV for crosschannel movement; the presence of Sb had only a little effect on diffusion of silver atoms. However it is worth noting that a reconstruction of the clean silver (110) surface was not observed in experiments. In 2006, the same year, Kong and Lewis [33] calculated the kinetic prefactor for diffusion of Ag atoms along the channels of the (110) plane using EAM potentials. The value for atom jumps turned out to be 2.9 × 10−3 cm2/sec when vibrational contributions were summed over all atoms of the crystal rather than using the local density approximation. For self-diffusion on Ag(110), Kim et al. [34] used molecular dynamics simulations and RGL interactions to arrive at a barrier of 0.277 eV for in-channel jumps, 0.388 eV for atom exchange, and 0.818 eV for cross-channel jumps, in good agreement with the work of Hontinfinde et al. [75]. In hetero-atom in-channel motion the barrier for diffusion of Ni was 0.268 eV, for Cu 0.249 eV, for Pd 0.337 eV, for Pt 0.492 eV, and 0.360 eV for Au. For Ag(110), experiments and reasonable calculations appear to be in sensible agreement, but most of the calculations were done using RGL potentials. There is also a lack of data about hetero-diffusion on these planes. No direct single atom investigations have been carried out, and that might have revealed a systematic error due, for example, to interactions. However, one thing seems to be clear from both experiment and theory – self-diffusion of silver proceeds mostly by simple hopping. The exchange mechanism is, however, likely to contribute at higher temperatures.

4.9

Tungsten: W(211) The (211) plane of tungsten is a surface intensively studied in diffusion experiments, but almost completely neglected by theory. Many different atoms have been examined on this plane, a model of which is given in Fig. 3.1 showing the rows which define the path of atoms moving over the surface. Note that the second layer in this model is not located symmetrically with respect to the first around ½0
11. The structure of this plane has been examined by low-energy electron diffraction [79], as well as by low-energy ion scattering [80]. These experiments have found that the first-to-second layer spacing is shortened by 9.3% (0.12 Å) and there is a lateral shift of 3.6% along the direction of the channels. However, it was shown recently that such asymmetry of the W(211) surface does not result in asymmetric diffusion [81]. To get an impression of how diffusion occurs, we will examine the available data.

4.9.1

Self-diffusion: FIM studies Pioneering work in diffusion studies [82] was carried out on W(211) with field ion microscopy in 1966, yielding rather odd results: a prefactor on the order of 10−7 cm2/sec or less, and a low activation energy of 0.56 eV, as in Fig. 4.28. These were the first ever direct measurements of atom movement on a surface. It may be that the presence of more

212

Diffusion on one-dimensional surfaces

Temparature T(K) 350

300

250

Mean-Square Displacement N02/t (Å2/sec)

1

(211)II

0.1

0.01

(321)II 0.001

3.0

Fig. 4.28

3.4 3.8 1000/T (K–1)

4.2

4.6

Arrhenius plots for the in-channel diffusivities of W atoms on W(211) and W(321) planes [82]. Deduced was a rather low value of 0.56 eV on W(211), and 0.87 eV for W(321); presence of many atoms on (211) may have caused low value.

than one atom on the plane interfered with the measurements. That may also have been true for the later studies by Bassett and Parsley [83] using the same technique; in 1970 they reported a diffusion barrier of 0.56 eV and another low prefactor of 3.8 × 10−7 cm2/sec. These were very early determinations, and later studies revealed more consistent behavior. Graham and Ehrlich [84] in 1975 reexamined the previous values of Ehrlich and Hudda [82] and from FIM observations reported a prefactor Do = 3 × 10−4 cm2/sec and an activation energy of 0.76 ± 0.07 eV. In a careful examination of this system by Flahive and Graham with an improved temperature calibration [85], an activation energy of 0.85 ± 0.05 eV and a prefactor of 1.5(× 10±1.5) × 10−2 cm2/sec was reported and is shown in Fig. 4.29. They also demonstrated that for planes with between 17–11 sites, the activation energy only barely depended on the size of the plane, but decreased between 11–9 sites, as indicated in Fig. 4.30. Here it should be noted that a reanalysis of Flahive and Graham’s data yielded the same barrier as before and a prefactor of 19.0(× 7.3±1) × 10−3 cm2/sec [86], shown in Fig. 4.31. These results are in reasonable agreement with the work of Wang [86], who examined a number of different adatoms on W(211); for tungsten adatoms he found a barrier of 0.82 ± 0.02 eV and a prefactor 7.7(× 1.9±1) × 10−3 cm2/sec. A comparison of the reanalyzed Flahive and Wang data is shown in Fig. 4.31. The value of the activation energy has turned out to be reproducible in further investigations with very good statistics. More recently diffusion on this plane has been studied by Senft looking for long jumps [87,88], whose results are given in Fig. 4.32. These were the first studies corrected for

213

4.9 Tungsten: W(211)

Temperature T(K) 300

280

260

3.0 ED = 0.85±0.05 eV Do = 1.5(× 10±1.5) × 10–2cm2/sec Mean-square displacement (Å/sec)

1.0

0.1

0.01 3.2

3.4

3.6

3.8

4.0

–1)

1000/T (K

Dependence of in-channel diffusivity for W adatoms on W(211) upon reciprocal temperature, determined by FIM (after Flahive and Graham [85]).

Activation Energy ED (kcal/mole)

Fig. 4.29

Self-Diffusion on W(211)

20.7

19.7

18.7

17

Fig. 4.30

15 13 11 Plane size (# sites)

9

Dependence of the activation energy determined for W(211) upon the number of sites on the plane. Measured energy is essentially constant for planes with 17 to 11 sites (after Flahive and Graham [85]).

diffusion occurring during temperature transients, while the sample is warming to the diffusion temperature and afterwards, during cooling. Her results, an activation energy of 0.81 ± 0.02 eV and a prefactor of 2.5(× 3.3±1) × 10−3 cm2/sec, fit in with the previous work of Wang and of Flahive and Graham, without problems even if diffusion during temperature transients were not taken into account in the previous studies. It is likely that in the temperature range investigated by Flahive and Graham transient motion was not important.

214

Diffusion on one-dimensional surfaces

Diffusivity D (× 10–16 cm2/sec)

10

260

Wang and Ehrlich ED = 0.82±0.02 eV Do = 7.7 (× 1.9±1) × 10–3cm2/sec

1

10–1 Flahive and Graham Reanalyzed

ED = 0.85±0.05 eV Do = 19(× 7.3±1) × 10–3cm2/sec

10–2

10–3 3.0

Fig. 4.31

Temperature T(K) 300 280

320

3.2

3.4 3.6 1000/T (K–1)

3.8

4.0

Comparison of different measurements of diffusion on W(211) [86]. Activation energies: 0.82 ± 0.02 eV for Wang, 0.85 ± 0.05 eV for Flahive and Graham [85].

Temperature T(K)

Diffusivity D (× 10–16cm2/sec)

10

320

310

300

290

270

Wang and Ehrlich ED = 0.82±0.02 eV Do = 7.7 ( × 1.9±1) × 10–3 cm2/sec

1

0.1 Senft and Ehrlich ED = 0.81±0.02 eV

Do = 2.5(× 3.3±1) × 10–3 cm2/sec

0.01

0.001

3

3.4 1000/T

Fig. 4.32

280

3.8 (K–1)

Comparison of W diffusivities on W(211) measured by Senft [88] and by Wang and Ehrlich [86].

215

4.9 Tungsten: W(211)

With the aim of exploring the mechanism of diffusion at high temperatures, Antczak [89] did more recent diffusion measurements over a 60 K range of temperatures and with huge statistics. An activation energy was found of 0.81 ± 0.02 eV and a prefactor of 3.41(× 2.40±1) × 10−3 cm2/sec, as shown in Fig. 2.19; this is in nice agreement with previous work. It should be noted that all the more recent studies give a prefactor of ~10−3 cm2/sec, which is expected if the effective vibrational frequency is ~1012 sec−1. Antczak observed the presence of longer as well as rebound transitions at around 300 K, and a gradual withdrawal of single jumps caused by longer transitions. The energetics of all kinds of observed long jumps were derived and data are presented in Chapter 3.3.4. She did not find any changes in diffusivity due to these alterations, and the mechanism of movement in self-diffusion up to 325 K on W(211) was established.

4.9.2

Self-diffusion: FEM studies At the same time as these FIM measurements, field electron microscopy was carried out by Tringides and Gomer [90]. In 1986 they measured self-diffusion on a (211) plane, relying on fluctuations of the field emission current through a probe hole in the form of a long slit. They found that diffusion, in the temperature range 568–694 K, occurred both along the channels, but also across them. Along the channels, the activation energy amounted to 0.57 eV, with a prefactor D o = 7.1 × 10−7 cm2/sec; across the channels, the figures were 0.27 eV and 1.8 × 10 −9 cm2/sec, as shown in Fig. 4.33. Even at the low temperature of 264 K, these figures imply that 10 jumps across the channels occur per second. However, cross-channel transitions have not been seen by field ion microscopy on this plane up to 340 K [89]. In the work of Tringides and Gomer [90], diffusion occurred with the tip biased negatively, and it is possible that the field induced cross-channel motion. Kellogg [91] in 1993 studied the effect of electric fields on the diffusion of platinum atoms on Pt(100), and found that a

Diffusivity D (cm2/sec)

In-Channel:

ED = 0.57eV Do = 7.1 × 10–7 cm2/sec

10–11

Cross-Channel:

ED = 0.27 eV Do = 1.8 × 10–9 cm2/sec

10–12

1.4

1.6

1.8

2.0

1000/T (K–1) Fig. 4.33

Dependence of in-channel and cross-channel self-diffusivity on W(211) upon reciprocal temperature, determined from fluctuations in field emission current (after Tringides and Gomer [90]).

216

Diffusion on one-dimensional surfaces

Diffusivity D (cm2/sec)

W(211) 10–10

10–11

10–12 ED = 0.56 eV Do = 4.8 × 10–7 cm2/sec 10–13 1.0 1.2 1.4

Fig. 4.34

1.6 1.8 2.0 1000/T (K–1)

2.2

2.4

Arrhenius plot for W atoms on W(211), deduced from field emission fluctuations (after Gong and Gomer [92]).

negative field promoted exchange diffusion, and that might be an explanation for these findings. As part of a larger study on thermal roughening of steps in 1988, Gong and Gomer [92] again looked at diffusion on the (211) plane of tungsten by measuring the fluctuations in the field emission current. They used a circular probe, so it was impossible to distinguish between in- and cross-channel movement. Their results, in Fig. 4.34, indicate a diffusion barrier of 0.56 eV and a prefactor Do = 4.8 × 10−7 cm2/sec. The authors suggested that the field applied to obtain field emission affected the activation energy. Choi and Gomer [93] extended these measurements in 1990; they covered a larger temperature range, from 500 K to more than 800 K, compared to the earlier work that extended from 613 K to 700 K. Their detector slit also was smaller. For diffusion along the channels they found an activation energy of 0.73 ± 0.02 eV and a prefactor Do = (3–2) × 10−5 cm2/sec. As shown in Fig 4.35a, there is a sharp drop in diffusion starting at a temperature of 826 K, but at 850 K the diffusivity rises again. In this last range, the activation energy proved to be 0.95 ± 0.02 eV with a prefactor of 5 × 10−5 cm2/sec. The results for diffusion across channels in the temperature range 526 K to 752 K yielded a barrier of 0.29 eV and a prefactor of 4.2 × 10−9 cm2/sec, shown in Fig. 4.35b, close to the earlier studies by Tringides and Gomer [90]. At temperatures above 752 K, cross-channel motion occurred over a barrier of 1.04 eV, with a prefactor of 5 × 10−4 cm2/sec. Choi and Gomer [93] interpreted the motion at lower temperatures as an exchange between an adatom with a lattice atom, and at high temperatures as diffusion over the channels. At the moment it is not really clear what is going on, but it is unlikely that single atom motion is being measured. More recently, Gong [94] has reported similar measurements, but with a round probe hole, which does not allow distinguishing in-channel and cross-channel diffusion. For the diffusion of tungsten on W(211) he found an activation energy of 0.56 eV and a prefactor Do = 4.8 × 10−7 cm2/sec, results and data exactly the same as in previous work [92]. The

217

4.9 Tungsten: W(211)

10–8

Temperature T(K) 1000 800 700 600 500 10–9 W/W(211) Diffusion along channels 752 K

10–10 W&E

Diffusivity D (cm2/sec)

Diffusivity D (cm2/sec)

10–9 10–10

10–11 T&G

10–11

(a) 10–12

Fig. 4.35

W/W(211) Diffusion across channels

T&G

1.0 1.2 1.4 1.6 1.8 2.0 2.2 1000/T (K–1)

(b) 10–12 1.0

1.2

1.4

1.6 1.8 2.0 1000/T (K–1)

2.2

2.4

Dependence of W diffusivity on W(211) surface upon reciprocal temperature. (a) Comparison of in-channel results from Choi and Gomer [93] with data from Tringides and Gomer [90] and with Wang and Ehrlich [86]. (b) Comparison of cross-channel measurements with the work of Tringides and Gomer [90] (after Choi and Gomer [93]).

FEM findings for in-channel motion agree with early field ion microscopic observations of diffusion on W(211), where several atoms might have interacted to produce such low values. Since with this technique there is no control in the experiments over the number of atoms on the plane, this is very likely the explanation for the data. Based on the lack of agreement between FIM and FEM data it is reasonable to conclude that FEM is not a technique really suitable for single atom self-diffusion studies. In FIM, movement of a single adatom can be followed directly, and the presence of a second adatom or defect can be detected. The electric field also does not influence the movement, since the state of the surface is frozen during imaging, so only thermal motion is detected.

4.9.3

Self-diffusion: theoretical studies There has been only little theoretical effort to estimate self-diffusion energetics on tungsten with many-body potentials. However, investigations with pair potentials started quite early. The first estimates of the surface migration barrier were done by Ehrlich and Kirk based on Morse potentials; they ranged from 0.78 eV for a realistic (fluctuating) surface, 1.09 eV for a surface with relaxed atomic positions, and 0.95 eV for a rigid surface [95]. Other early calculations were made using Morse function interactions by Wynblatt and Gjostein [96];

218

Diffusion on one-dimensional surfaces

they arrived at a barrier of 0.432 eV for jumps, considerably below the current experimental values. Later studies were done by Flahive and Graham [20]. Using Morse potentials again, they estimated a self-diffusion barrier of 0.28 eV for in-channel motion on a fully relaxed surface, and 0.91 eV for an unrelaxed surface, with the data presented in Fig 4.9. Values on the relaxed surface obtained from Morse potentials were much too low compared with experiments. Lennard-Jones potentials were used by Doll and McDowell [97] in 1982 and from their calculation they arrived at an activation energy of 0.887 ± 0.224 eV. In 1984 Voter and Doll [98] continued studies with Lennard-Jones potentials and found a barrier of 0.952 ± 0.182 eV and a prefactor of 2.8 × 10−3 cm2/sec using Monte Carlo TST; from simple TST they obtained 0.822 eV and a prefactor of 1.6 × 10−3 cm2/sec. Values obtained from L-J potentials are not too far from the experimental data. The only work so far with many-body potentials was done in 1994. Xu and Adams [99] did work on tungsten using the fourth-moment approximation to tight-binding theory. For in-channel diffusion they calculated an activation energy of 0.79 eV with a prefactor of 1.41 × 10−3 cm2/sec; for cross-channel diffusion by exchange their activation energy turned out quite high, 2.0 eV with a prefactor of 9.4 × 10−4 cm2/sec. The estimate for in-channel diffusion is quite close to the results from experiments.

4.9.4

Hetero-diffusion: nickel, molybdenum, rhodium Diffusion of nickel was examined in FIM studies on W(211) by Senft [88], who found an activation energy of 0.46 ± 0.01 eV and a prefactor of 1.9(× 2.7±1) × 10−3 cm2/sec, as shown in Fig. 4.36. This study was quite careful and statistics were quite reasonable, so it is probably close to the real value for Ni. There are also indications of long jumps in diffusion at 179 K, as discussed in Chapter 3. The first examination of molybdenum diffusion on W(211) was done by Bassett and Parsley [83] in 1970 with a few atoms on the plane using FIM. They reported an activation

200 10

180

Temperature T (K) 160

140

Diffusivity D (× 10–16 cm2/sec)

Ni on W(211) 1

10–1

10–2

10–3

Do = 1.9(× 2.7±1) × 10–3 cm2/sec 10–4 5.0

Fig. 4.36

ED = 0.46±0.01 eV

5.5

6.0 6.5 1000/T (K–1)

7.0

7.5

Arrhenius plot for in-channel diffusion of Ni on W(211) (after Senft [88]).

219

4.9 Tungsten: W(211)

10 Single Adatom on W(211) Atom Pair on W(211)

Mean-Square Displacement (Å/sec)

Single Adatom on W(321)

1

0.1

0.01

3.0

3.5

4.0

4.5

–1)

1000/T (K Fig. 4.37

Dependence of diffusivity for Mo on W(211) and W(321) upon reciprocal temperature. For Mo atoms on W(211) a barrier of 0.57 eV was obtained in observations with FIM and 0.55 eV for W(321) (after Sakata and Nakamura [100]).

energy for diffusion of 0.55 eV, with prefactor of 2.4 × 10−6 cm2/sec. In 1975, Sakata and Nakamura [100] carried out a variety of studies on W(211) also in an FIM, and for the diffusion of Mo atoms they found an activation energy of 0.57 eV, with a prefactor of 9.3 × 10−7 cm2/sec. These results were based on only ~ 20 measurements at each of five temperatures, and are shown in Fig. 4.37. The agreement with the earlier work is very good, both in the low activation barrier and the prefactor, but this low value is open to considerable doubt. Wang [86] later again used field ion microscopy and studied diffusion at nine temperatures, with one hundred observations at each, and found quite a different result: an activation energy of 0.71 ± 0.01 eV, and a prefactor of 2.0(× 1.6 ±1) × 10−3 cm2/sec. The disagreement with earlier work is significant, and the latest measurement appears more reliable, with better statistics and with only a single atom on the plane. The movement of Mo atoms at 262 K proceeded by single jumps only [101] for the investigated temperature range, as is clear from Fig. 3.85. Diffusion of rhodium on the (211) plane of tungsten was examined by Wang [86] in the FIM. Results obtained at ten temperatures gave an activation energy of 0.536 ± 0.001 eV and a prefactor of 3.3(× 1.5±1) × 10−3 cm2/sec, derived from the Arrhenius plot in Fig. 4.38. Investigations by Wang et al. [101] at temperatures of 189 K and 192 K gave negligible amounts of double transitions, and higher temperatures were not investigated. No other studies have been reported for this system.

220

Diffusion on one-dimensional surfaces

102

Diffusivity D (× 10–16 cm2/sec)

Re

W Mo Ir

200

Rh

180

ED eV

0.536±0.001

10

0.67±0.01 1

0.71±0.01 0.82±0.02 10–1 10–2

0.83±0.01

Do

× 103

3.3 (× 1.5±1) 0.61 (× 1.5±1)

cm2/sec

0.73 (× 1.6±1) 7.7 (×1.9±1) 2.0 (× 1.6±1)

10–3

2.8

Fig. 4.38

Temperature T(K) 280 240

320

360

3.2

3.6

4.0 4.4 1000/T (K–1)

4.8

5.2

5.6

Dependence of adatom in-channel diffusion on W(211) upon reciprocal temperature, obtained from FIM measurements of atom displacements [86]. Uncertainties derived by simulation. Temperature T (K)

Diffusivity D (× 10–16 cm2/sec)

102

150 140

130

120

110

100

Pd on W(211)

10

1 10–1 10–2

10–3 10–4 6.5

ED = 0.32±0.01 eV Do = 3.9 (× 2.4±1) × 10–4 cm2/sec 7

7.5

8.5 8 9 1000/T (K–1)

9.5

10 10.5

Fig. 4.39

Arrhenius plot for in-channel diffusion of Pd on W(211), obtained from observations in FIM (after Senft [88]).

4.9.5

Hetero-diffusion: palladium, rhenium, iridium Senft [87,102] utilized the FIM to examine the movement of palladium, as shown in Fig. 4.39, and reported a barrier of 0.32 ± 0.01 eV and a prefactor of 3.9(× 2.4±1) × 10−4 cm2/sec. She found a significant number of double and triple jumps for this system at a temperature of 133 K, as described in detail in Chapter 3. It should be noted that despite differences in the activation energies for the different

221

4.9 Tungsten: W(211)

–35

Pd/W(111) ED = 1.02±0.06 eV Do = 1 × 10–3 cm2/sec

–36

In D

–37

Pd/W(211) Pd/W(110) ED = 0.32±0.02 eV ED = 0.51±0.03 eV –4±0.8 Do = 1.4 × 10–3±1.9 cm2/sec Do = 5.1 × 10 cm2/sec

–38 –39 –40 –41 –42

2

3

4

5

6

7

8

9

10

1000/T (K–1) Fig. 4.40

Diffusion of palladium on W(211), W(111), and W(110), shown in Arrhenius plots obtained from FIM observations (after Fu et al. [103]).

types of jumps, the prefactor in the diffusivity is still around 10−3 cm2/sec. In 2002, Fu et al. [103] looked at diffusion of palladium on tungsten surfaces also using FIM, as shown in Fig. 4.40. Movement on W(211) was measured at four temperatures, and an activation energy for diffusion of 0.32 ± 0.02 eV was found, with a prefactor of 5.1 × 10−4.0 ± 0.8 cm2/sec, in excellent agreement with the results of Senft. Movement of Re has been the first adatom probed by Bassett and Parsley on W(211) [104] in this multi-atom study in an FIM. They found an activation energy for diffusion of 0.88 eV, and a prefactor of 1.1 × 10−2 cm2/sec. They also checked the effect of an electric field of 0.9 V/Å on the diffusivity and observed an increase of 0.009 eV in the activation energy and 0.7 × 10−2 cm2/sec in the prefactor, changes that appeared insignificant. However, observations were done with a few atoms present on the surface. Subsequent work was reported by Stolt et al. [105] in 1976, and later still by Wang [86] in 1988, both with field ion microscopy. With approximately 100 observation per temperature in the range 290 K to 351 K, Stolt et al. found an activation energy of 0.86 ± 0.03 eV and a prefactor of 2.2(× 2.8±1) × 10−3 cm2/sec from the Arrhenius plot in Fig. 4.41, whereas Wang reported an activation energy of 0.83 ± 0.01 eV and a prefactor of 0.73(× 1.6±1) × 10−3 cm2/sec. These measurements, done without “zero-time” compensation, are in reasonable agreement. It should be noted that at these diffusion temperatures, atom jumps during temperature transients may have been negligible. Analysis of the distribution of displacements for temperatures 300 and 317 K did not detect the presence of double jumps [101]. The first study, with many atoms present on the plane, seems not to lie too far from later measurements with only one atom, which might indicate weak interactions between the atoms present in different channels. Bassett and Parsley [83] studied the motion of iridium on W(211) with several atoms on the surface using the FIM technique; they found an activation energy of 0.58 eV and a prefactor of 2.7 × 10−5 cm2/sec. In a brief comparison study of atom and dimer diffusion, Reed [106] reported an activation energy of 0.53 ± 0.05 eV with a prefactor of 5 × 10−7 cm2/sec for single adatoms, also measured by FIM, as shown in Fig. 4.42. These values appear too low. A rather higher activation energy of 0.67 ± 0.01 eV and

222

Diffusion on one-dimensional surfaces

350

Diffusivity D (cm2/sec)

102

Temperature T(K) 325 300

10

1

10–1

Re on W(211) ED = 0.86±0.03 eV Do = 2.2(× 2.8±1)× 10–3 cm2/sec

10–2 2.6

2.8

3.0

3.2

3.4

3.6

1000/T (K–1) Fig. 4.41

Arrhenius plot for in-channel diffusivity of Re atom on W(211), derived from FIM measurements [105].

Mean-Square Displacement (Å2/sec)

1.0

Temperature T(K) 250

0.1

0.01 Iridium Dimers ED = 0.67 ± 0.06 eV Do = 9 × 10–6 cm2/sec Single Atoms ED = 0.53 ± 0.05 eV Do = 5 × 10–7 cm2/sec 0.001 3.2

Fig. 4.42

300

3.4

3.6

3.8 4.0 4.2 1000/T (K–1)

4.4

4.6

Dependence of Ir diffusivity on W(211) upon reciprocal temperature [106]. Results useful only for comparison of atom and dimer behavior, not for actual kinetics.

223

4.9 Tungsten: W(211)

prefactor of 0.61(× 1.5±1) × 10−3 cm2/sec was found by Wang [86] in extensive experiments. These utilized the FIM and were shown previously in Fig. 4.38, which involved roughly 100 observations at each of eleven temperatures. The earlier studies by Reed [106] are useful comparing single atom and dimer motion, but should be ignored in examining details of single atom diffusion. Investigation of the type of jumps Ir atoms make at temperatures of 250 K, 261 K, and 272 K, showed a small contribution of double jumps. The ratio of double to single jumps was 0.004, 0.044 and 0.027, too low to claim they were significant [101], however, still missing are “zero-time” corrections.

4.9.6

Hetero-diffusion: miscellaneous adatoms There are a number of studies which have concentrated on only one or two different adatoms; these are grouped together here. In 1970, Bassett and Parsley [107] examined diffusion of tantalum on W(211) in the FIM. They reported a diffusion barrier of 0.48 eV and a prefactor of 0.9 × 10−7 cm2/sec. These values are undoubtedly too low, possibly influenced by interactions with other adatoms, and should be reexamined. Field emission measurements of gold adsorption on W(211) were made quite early by Jones and Jones [108] in 1976. They measured the rate at which gold swept over this plane, and derived an activation energy of 0.19 eV. They arrived at this value at a high coverage, half of a monolayer, and it is clear that information about single atom diffusion was not generated. A different type of diffusion study was carried out by Bayat and Wassmuth [109], who examined the movement of potassium on W(211) with a surface ionization microscope. The amount of potassium on the surface was extremely small, less than 10−3 of a monolayer, so these measurements should correspond to the limit of zero concentration. Studies were done for quite high temperatures from 840 K to 1400 K, and not only diffusion but also evaporation took place. The latter was examined separately, using the relation for the lifetime τ on the surface of τ ¼ τ 0 expðQi =kTÞ;

(4:6)

for Qi, the desorption energy of potassium ions, a value of 2.52 ± 0.04 eV was found from the data in Fig. 4.43a. The root mean-square displacements along and across the channels were given as rffiffiffiffiffiffiffiffiffiffi D Effi x2k ¼ ðxk Þ exp½ðQi
Ek Þ=2kT; (4:7) qffiffiffiffiffiffiffiffiffiffi  2 x? ¼ ðx? Þ exp½ðQi
E? Þ=2kT: The studies of diffusion are shown in Fig. 4.43b, and gave a barrier to in-channel motion þ0:10 eV. The preof Ek ¼ 0:46  0:08 eV and to cross-channel motion of E? ¼ 0:76
0:08 2 factor for both processes proved to be roughly the same, 0.3 cm /sec. Diffusion of lithium on W(211) was examined by Biernat et al. [110] who measured fluctuations in the field electron emission current. They looked at coverages starting from 0.4 ML, and observed a strong coverage dependence of the activation energy as well as the prefactor. For 0.4 ML, they found an activation energy in the

224

(a)

Diffusion on one-dimensional surfaces

1000

Temperature T (K) 900

800 (b)

102

Temperature T(K) 1400

1100

1000

900

500

K/W(211) 101

200

K/W(211)

– -– (X[111] = Xll

– – (X[110] = X⊥

50 – X (μm)

Lifetime τ (sec)

1200

100

Ql-Ell = 2.08 eV – (Xll)0 = 6 × 10–4μm

20

10 Qi = 2.52 ± 0.04 eV

τ0 = 1.2 × 10–14 sec

10–1

Qi -E⊥ = 1.78 eV – (X⊥)0 = 6 × 10–4μm

5

0.7 10–2

Fig. 4.43

1.0

1.1 1000/T (K–1)

0.8

1.0 0.9 1000/T (K–1)

1.1

1.2

1.2

(a) Lifetime of potassium on W(211) as a function of reciprocal temperature, derived from surface ionization microscopy. (b) Arrhenius plot for in- and cross-channel diffusion of K on W(211), giving a barrier of 0.46 ± 0.08 eV for in-channel motion and 0:76þ0:10
0:08 for cross-channel diffusion (after Bayat and Wassmuth [109]).

direction equal to 0.49 eV, with a prefactor of 2.0 × 10−3 cm2/sec. At low temperature the movement was strictly one-dimensional, at higher temperature two-dimensional movement became apparent. At a coverage of 0.6 ML, they obtained an in-channel barrier of 0.46 eV and a prefactor of 1.2 × 10−3 cm2/sec, as well as a cross-channel activation energy of 0.54 eV and a prefactor of 5.8 × 10−3 cm2/sec. However, concentrations were so high that diffusion for single atoms was strongly affected by interactions. Strontium on W(211) was investigated in Naumovets’ group by monitoring changes in work function during spreading over the surface [111]. Coverages down to about 0.05 ML were studied, and the activation energies for diffusion are shown in Fig. 4.44, with a barrier of 0.87 eV in the most dilute state. Unfortunately this technique cannot control defects at the surface, which might influence the diffusion barrier and are likely to be responsible for the high prefactor of ~0.4 cm2/sec. In 1999, Loburets [59] studied the diffusion of copper on W(211) by measuring work function changes. Regrettably his lowest concentrations of Cu were ~1ML, so that no information about individual atom behavior could be derived from his work. Although it would be desirable to have more data for hetero-atom diffusion, the experiments on single adatoms are in reasonably good shape.

225

ED (eV)

4.10 Tungsten: W(321)

C-I

PT1 T > 745 K

2

T < 715 K

1

1.3

0

Iog Do

1.1

log Do(cm2/sec)

1.2 ED

1.0 0.9 0.0

0.1

0.2

0.3 0.4 Coverage Θ

0.5

0.6

Fig. 4.44

Diffusion characteristics for strontium on W(211) at different surface coverages (after Loburets et al. [111]).

4.10

Tungsten: W(321)

4.10.1

Self-diffusion Much less has been done to characterize diffusion on the (321) plane of tungsten, a schematic of which was shown in Fig. 3.1. What is significant here is that this plane is again made up of channels; however, they are wider and deeper than on W(211). All studies except for those by Gong and Gomer [92] utilized field ion microscopy. The first measurements were done by Ehrlich and Hudda [82] in 1966. For tungsten adatoms migrating on W(321) they found a barrier of 0.87 eV and a prefactor of 1 × 10−3 cm2/sec; their data are shown in Fig. 4.28. It should be noted that these measurements were done with a few atoms on the plane. In 1970, Bassett and Parsley [107] examined the mobility of tantalum, tungsten, and rhenium, again with multi-atom deposits. For tungsten they arrived at a barrier of 0.84 eV and a prefactor of 1.2 × 10−3 cm2/sec. Graham [84] reported new determinations for self-diffusion, with a barrier of 0.81 ± 0.078 eV and a prefactor of 1 × 10−4 cm2/sec, but this time with only one atom on the plane giving more reliable results. Some observations of tungsten adatoms were reported by Nishigaki and Nakamura [112], who made a few observations of in-channel self-diffusion at 330 K, and found an activation energy of 0.89 eV. In 1980 Flahive and Graham [85] explored shallow and deep sites on W(321) shown in Fig. 3.11. For movement from shallow to deep sites they found a barrier of 0.65 eV, assuming a standard prefactor; for motion along the deep sites the barrier was ~0.87 eV. Gong and Gomer [92] in 1988 looked at self-diffusion on the W(321) plane, measuring fluctuations in the field electron emission current; results, including activation energies, are given in Fig. 4.45. The low values for the prefactors are highly suspicious. Fast diffusion was two-dimensional, slow diffusion was one-dimensional. It shows once more that FEM is really not a suitable tool for diffusion measurement of single atoms. First calculations for diffusion on this plane were done by Ehrlich and Kirk [95] in 1968. Using Morse potentials they found energy barriers of 0.96 eV, 1.09 eV, and 0.91 eV

226

Diffusion on one-dimensional surfaces

10–9

ED = 0.71 eV

Diffusivity D(cm2/sec)

Do = 1.4 × 10–5 cm2/sec 10–8

W (321) ED = 0.35 eV Do = 1.8 × 10–8 cm2/sec

10–7

ED = 0.58 eV Do = 7.3 × 10–12 cm2/sec

10–6

0.8

1.2

1.6

2.0

2.2

1000/T (K–1) Fig. 4.45

Arrhenius plots for W atoms on W(321) derived from measurements of fluctuations in field emission current. Fast diffusion with barriers of 0.35 eV and 0.706 eV is two-dimensional (after Gong and Gomer [92]).

for rigid, relaxed, and fluctuating surfaces respectively, quite close to the experimental results. Early theoretical estimates for tungsten diffusion on W(321) were made with Morse potentials by Flahive and Graham [20]. For in-channel movement they calculated a barrier of 0.13 eVon a fully relaxed surface, much lower than what was found by experiment; on a surface with no relaxation the barrier increased to 1.07 eV. Xu and Adams [99] much later, in 1994, used interactions derived from tight-binding theory by the fourth-moment method. For in-channel movement by hopping between deep sites they estimated a barrier of 0.73 eV and a prefactor of 1.36 × 10−3 cm2/sec. Other transitions, between shallow sites and from deep to shallow sites required appreciably more energy. However, atom exchange in moving from shallow to deep sites, shown in Fig. 3.11, involved an activation energy of only 0.23 eV and a prefactor of 3.4 × 10−4 cm2/sec. The in-channel barrier is not too far from experiments. The exchange from shallow to deep sites predicted here was not observed experimentally by Flahive and Graham [20]. Agreement of the diffusivities in the different measurements of the diffusion of tungsten atoms (except for the field emission studies) is surprisingly good, considering that the early work was done in different laboratories with different equipment. Even values measured with a few atoms on the plane do not stand out, probably due to negligible interactions for adatoms separated by a channel of the plane. Comparison of these findings with results for tungsten on W(211) is interesting, since activation energies and prefactors are quite close. The theoretical estimates with many-body potentials are definitely in order for this surface.

4.10.2

Hetero-diffusion In 1969 Bassett and Parsley [104] looked at the diffusion of Re atoms on W(321) in the FIM, and found an activation energy of 0.88 eV and a prefactor of 0.5 × 10−3 cm2/sec.

4.11 Iridium: Ir(110), (311), (331)

227

They also tested the effect of an electric field on the diffusion. With a field of 0.9 V/Å they noted an increase of 0.06 eV in the activation barrier and 1.5 × 10−3 cm2/sec for the prefactor, changes that did not appear significant. A year later, Bassett and Parsley [107] examined the mobility of tantalum, tungsten, and rhenium on this plane, again with multi-atom deposits. For Ta they reported a diffusion barrier of 0.67 eV and a prefactor of 1.9 × 10−5 cm2/sec. For rhenium adatoms their results were a diffusion barrier of 0.88 eV and a prefactor of 4.8 × 10−4 cm2/sec. These prefactors appear somewhat low, possibly influenced by interactions with other adatoms, and should be looked at again. Sakata and Nakamura [100] examined diffusion of molybdenum adatoms on W(321) also in an FIM, for which they gave an activation energy of 0.55 eV and a prefactor Do = 1.2 × 10−7 cm2/sec, as shown in the Arrhenius plot in Fig. 4.37. This is orders of magnitude below the usual prefactor values. Only one atom was on the plane in these experiments, but the statistics were rather limited. From this survey it is clear that there is no reliable information about hetero-atom movement on the W(321) plane.

4.11

Iridium: Ir(110), (311), (331) On Ir(110), Wrigley [113] in 1980 used the Atom Probe to establish that in a majority of the observations, motion of iridium atoms occurred across the rows rather than just along them. He also looked at the temperature dependence of the cross-channel diffusion of iridium atoms [114], and reported an Arrhenius plot, shown in Fig. 3.45; this yielded an activation energy of 0.74 ± 0.09 eV and Do = 1.4(×16±1) ×10−6 cm2/sec. Measurements were done at only five temperatures, and in view of the low value of the prefactor must be considered with caution. He also looked at tungsten atoms moving over the iridium surface and using the Atom Probe proved that the exchange mechanism took place on this plane; he did not examine the energetics of this movement, however. Eleven years later, Chen and Tsong [41,115] using field ion microscopy reported detailed self-diffusion measurements on a field evaporated Ir(110) surface, just as above, but also on the (1 × 2) reconstructed plane. Their results are displayed in Fig. 3.49 and 4.46. On the simple Ir(110) plane, the activation energy for self-diffusion by in-channel motion was 0.80 ± 0.04 eV, with a prefactor of 4 × 10−3 ± 0.8 cm2/sec. For cross-channel motion an activation energy of 0.71 ± 0.02 eV was found, as in Fig. 3.49. This energy, derived from measurements at only four temperatures, is in agreement with the previous results of Wrigley. The prefactor of 6 × 10−2 ± 1.8 cm2/sec is much more sensible, however. On the reconstructed (110) plane, with much deeper channels, diffusion occurred only along the direction and over a higher barrier, 0.86 ± 0.03 eV, with a prefactor of 1.2 × 10−3±1 cm2/sec. There is the possibility of a metastable walk on this surface, but the authors did not explore the mechanism of movement. Chen et al. [116], again using the FIM, did studies of rhenium atoms on Ir(110) and Ir(311). On the latter plane, rhenium atoms diffused along the channels above 250 K and

228

Diffusion on one-dimensional surfaces

0.0

–0.5

ED = 0.86 ± 0.03 eV Do = 1.2 × 10–3±1 cm2/sec

In N

–1.0

–1.5

–2.0 Ir on (1 × 2)Ir(110) –2.5 3.0 Fig. 4.46

3.1 3.2 1000/T (K–1)

3.3

Arrhenius plot for diffusivity of Ir on Ir(110)-(1 × 2) plane, giving higher activation energy than on (1 × 1) plane (after Chen and Tsong [41]).

did not undergo any atom exchanges. On Ir(110), after depositing a rhenium atom and heating to 256 K, there clearly was an exchange with an atom from the channel walls, and an iridium atom appeared in the neighboring channel. Rhenium atoms did not migrate along the channels; the energetics of the processes were not measured, however. In 1996, Fu et al. [117] examined self-diffusion on Ir(311) and Ir(331) planes with the FIM. The activation energy found for the first was 0.72 ± 0.02 eV, with a prefactor of 1 × 10−3±0.4 cm2/ sec. On the (331) plane, the energy barrier, derived from the Arrhenius plot in Fig. 4.47, was 0.91 ± 0.03 eV with a prefactor of 1 × 10−1.9±0.5 cm2/sec. Movement of the adatom was one-dimensional. A number of calculations have been made for the self-diffusion of iridium. That started with the calculations of Flahive and Graham [20] using Morse potentials, with the results shown in Fig. 4.9. For Ir(311) they got a value around 0.6 eV, for Ir(110) ~0.75 eVand for Ir(331) ~0.95 eV. Shiang et al. [118] worked with both EAM and RGL interaction potentials. They found an activation energy of 0.92 eV for exchange with EAM potentials and 0.75 eV with RGL, quite a big difference between values from these two methods. For diffusion by atom jumps along the channels, EAM gave an activation energy of 0.70 eV and with RGL 0.83 eV. Jumps from one channel to an adjacent one were found to require 2.58 eV in calculations using the RGL approximation. It turns out that EAM predicts a lower value of the activation energy for jumps compared with exchange, while RGL predicts the opposite sequence. For both methods, however, the differences in activation energies for both mechanisms are not that big, indicating coexistence of both mechanisms, in agreement with experiment. In 1996, Chang et al. [119] continued with embedded atom calculations for single atom self-diffusion on Ir(110). They found an activation energy of 0.70 eV for hopping along the channels, while exchange in both directions and occurred over the same barrier of 0.81 eV, results not too far from experiments. Similar to the previous

229

4.11 Iridium: Ir(110), (311), (331)

–37

In D

–38

–39

Ir(311)

Ir(331) –40 ED = 0.91 ± 0.03 eV Do = 1 × 10

3.0

–1.9±0.5

ED = 0.72 ± 0.02 eV Do = 1 × 10–3±0.4 cm2/sec

2

cm /sec

3.2

3.4 1000/T

Fig. 4.47

3.6

3.8

4.0

(K–1)

Dependence of diffusivity for Ir atoms on Ir(331) and Ir(311), obtained in observations with FIM (after Fu et al. [117]).

EAM study [118] hopping seemed to be slightly more favorable, opposite to what was observed in experiment, where exchange was a bit more frequent event. Hopping into the adjacent channel required at least 2.5 eV. The lack of a preference for the direction of exchange does not agree with the experimental findings of Chen et al. [41] for the same system. Agrawal et al. [7] utilized Lennard-Jones potentials to calculate the barrier to jumps of iridium atoms on Ir(110). For cross-channel jumps, they estimated a very high activation energy of 2.47 eV and a prefactor of 1.1 × 10−2 cm2/sec, but for in-channel transitions the barrier was very high, 1.48 eV, with a prefactor 4.7 × 10−3 cm2/sec. No investigations of atomic exchange were performed. Calculations with more reliable RGL potentials were done by Ndongmouo and Hontinfinde [32], who came up with a barrier of 1.22 eV for diffusion along the (1 × 2) reconstructed plane; for the (1 × 1) surface, the barrier was lower, 1.15 eV, but still high. Cross-channel exchange events were estimated from the surface energy to occur over a barrier of 1.60 eV, and jumps between adjacent channels required an activation barrier of 2.21 eV. All values are much higher than the experimental data as well as previous estimates with many-body potentials. The experimental results for self-diffusion on Ir(110) seem to be reasonable. However, the calculations, even when done for the same interactions, do not appear reproducible. The work of Shiang [118] with RGL interactions is close to agreement with experiments, the studies by Ndongmouo and Hontinfinde [32], with the same potentials, are far away. Only Agrawal et al. [7] made an effort to provide prefactors for the diffusivity, but all their data stand apart from experimental findings, probably as they included only pair interactions. Theory as well as experiment are in agreement about the coexistence of two mechanisms: hopping and exchange.

230

Diffusion on one-dimensional surfaces

4.12

Platinum: Pt(110), (311), (331)

4.12.1

Experimental data Bassett and Webber [120] were the first to study diffusion on platinum surfaces, studies which gave very interesting and important results. Using field ion microscopy, FIM, on the (110) surface, they found in 1978 that atom movement occurred along the direction of the channels, as expected, with an activation energy of 0.84 ± 0.1 eV and a prefactor of 8 × 10−3 cm2/sec. The surprise, however, was that diffusion also occurred across the channels, with a lower barrier of 0.78 ± 0.1 eV and a prefactor of 1 × 10−3 cm2/sec. They also did calculations using Morse potentials and obtained 0.64 eV for the inchannel diffusion barrier and 1.97 for inter-channel jumps. We can see that Morse potentials are not very useful for explaining cross-channel diffusion. However, this study for the first time ever showed the existence of a cross-channel mechanism in diffusion on the surface. In addition, studies were made of platinum diffusion on the channeled (311) and (331) planes. On these, motion always occurred only along the channels. On the (311), the diffusivity was obtained by assuming a prefactor of ðkT=hÞ‘2 ; this gave an activation energy of 0.69 ± 0.2 eV. On the (331) plane the activation energy was higher, 0.84 ± 0.1 eV, with a prefactor of 4 × 10−4 cm2/sec. From calculations with Morse potentials they obtained 0.50 eV for the (311) barrier and 0.77 eV for diffusion on the (331) plane. The values are of the same order as the experiments for these planes. Measurements of platinum motion on Pt(110) repeated eight years later by Kellogg [121] for both in-channel as well as cross-channel movement in the FIM gave similar results. For the former he found a barrier of 0.72 ± 0.07 eV and a prefactor of 6 × 10−4; for cross-channel motion these values were 0.69 ± 0.07 eV and 3 × 10−4 cm2/sec. As already shown in Fig. 3.47, Arrhenius plots were based on only four temperatures, and within the limits of the experiments in-channel and cross-channel jumps occurred at the same rate, what meant that on this plane both mechanisms coexisted. Kellogg [122] in 1986 also looked at self-diffusion on the platinum (311) plane with the FIM. He reported a diffusion barrier of 0.60 ± 0.03 eV and a prefactor of 1.9 × 10−4 cm2/sec, as shown in Fig. 4.48, in reasonable agreement with the results of Bassett and Webber [120]. No indication of cross-channel diffusion was found, only in-channel motion occurred. We do want to mention the work of Preuss et al. [123], who did very interesting studies of self-diffusion on single crystal Pt(110), although they did not measure single atom motion. They looked at the decay of a platinum deposit by analyzing surface profiles occurring by diffusion at temperatures above 1200 K, and as shown in Fig. 4.49, came to the conclusion that diffusion occurred both along the surface channels and across them. The activation energy for movement along the channels proved to be 1.70 eV and across the channels 3.2 eV, much above the results from single atom experiments. A few results for other adatoms have been reported. Bassett and Webber [120], again using field ion microscopy, estimated a barrier of 0.80 ± 0.15 eV for diffusion of iridium along channels on Pt(110), and the same value, 0.80 ± 0.15 eV for cross-channel motion. These figures were estimated assuming a value for the prefactor. Morse potential

231

4.12 Platinum: Pt(110), (311), (331)

In<x 2/t > (<x 2/t> in cm2/sec)

–33

–35 –37 –39 ED = 0.60 eV Do = 1.9 × 10–4 cm2/sec

–41 –43 3.5

Fig. 4.48

3.8

4.1 4.4 1000/T (K–1)

4.7

5.0

Arrhenius plot for the diffusivity of Pt on Pt(311) plane examined in FIM (after Kellogg [122]). Temperature T(K) 2042 1600 1200

10–4

Diffusivity D(cm2/sec)

Pt(110)

10–5

[110] ED = 1.7 eV Do = 2.9 cm2/sec

10–6

10–7

10–8

Fig. 4.49

Tm

[001] ED = 3.2 eV Do = 4 × 104 cm2/sec 4

5

6 7 8 9 10 000/T (K–1)

10

11

Dependence of mass-transfer diffusivity of Pt on Pt(110) upon reciprocal temperature for in-channel [1
10] and cross-channel [001] motion (after Preuss et al. [123]).

calculations for the same system predicted a barrier of 0.66 eV for jumps along the channels. On the (311) plane the activation energy from experiments was 0.74 ± 0.15 eV, from calculations a value of 0.54 eV was derived. On Pt(331), movement of iridium required overcoming a barrier of 0.81 eV from calculations, and there were no experimental data derived. For gold on Pt(110), diffusion occurred along the channels over a barrier of 0.63 ± 0.15 eV, again estimated by assuming a prefactor. Calculations with Morse potential gave a barrier of 0.57 eV. What is worth noting is that gold behaves quite differently than Ir or Pt on Pt (110): for gold only one-dimensional movement was observed on this

232

Diffusion on one-dimensional surfaces

In[<x 2>/t (cm2/sec)]

–34

–36 –37 –38 –39 –40 4.0

Fig. 4.50

Pd on Pt(110)

–35

ED = 0.58 ± 0.05 eV Do = 3.5 × (10±1.2) × 10–3 cm2/sec 4.2

4.4 4.6 1000/T (K–1)

4.8

5.0

Arrhenius plot for the diffusion of Pd atom on Pt(110) plane, determined by FIM (after Kellogg [124]).

plane. On Pt(311), the barrier measured for gold movement was 0.56 ± 0.1 eV, and a prefactor of 3 × 10−1 cm2/sec while calculations predicted 0.41eV. On Pt(331), according to calculations, movement of gold required overcoming a barrier of 0.68 eV. Field ion microscope studies of palladium atoms diffusing on the Pt(110) plane were made by Kellogg [124]. As shown in Fig. 4.50, in the temperature range 207–235 K he found an activation energy of 0.58 ± 0.05 eV and a prefactor of 3.5 × (10±1.2) × 10−3 cm2/ sec for atoms hopping along the channels. Unfortunately, the statistics were limited, resulting in significant scatter of the data. Palladium is similar to gold in moving in only one dimension, and only by jumps, not by exchange. Linderoth et al. [125,126] in 1997 examined self-diffusion on the Pt(110)-(1 × 2) plane, shown in Fig. 3.93; it is a Pt (110) surface with every second row of platinum removed. They observed atomic jumps using the STM, with a tunnel junction resistance of 100 MΩ, at which the effect of the probing tip on the adatoms was negligible. Diffusion here occurred entirely along the channels forming this plane, and from measurements of the distribution of displacements, in Fig. 3.92, it was possible to establish that in addition to jumps between nearest-neighbor positions, longer transitions also occurred. From an Arrhenius plot of the jump rates, shown in Fig. 3.93, they were able to arrive at an activation energy of 0.81 ± 0.01 eV with a frequency prefactor of 1010.7 ± 0.2 sec−1 for single jumps. The situation with what were called “double” jumps, which had an activation energy of 0.89 ± 0.06 eV and a frequency factor of 1010.9±0.8 sec−1, is more complicated, as these turned out not to be just double length transitions along the channel bottoms. A year after this work, Montalenti and Ferrando [127,128] worked on the missing row reconstructed Au(110) plane, doing molecular dynamics simulations with energetics from RGL potentials. They found that the adatom is not confined to the bottom of the trough, but can instead run up the channel sides and diffuse rapidly there until it returns again to the bottom. The implication of these findings are clear – such diffusion should also occur on the reconstructed Pt(110) plane. The next year, in 1999, Lorensen et al. [129] came out with density-functional calculations for diffusion on this surface, shown in Fig. 4.51. They found that in addition

233

4.12 Platinum: Pt(110), (311), (331)

Energy E(eV)

1.0

Facet II

Facet I

ΔE

HCP

HCP FCC

0.5

0 Reaction Coordinate Fig. 4.51

Potential diagram for Pt diffusion on Pt(110) plane calculated with density-functional method in generalized gradient approximation. Dashed curve is for direct jump along channel (after Lorensen et al. [129]).

to atoms following a direct path along the bottoms of the troughs, platinum atoms could indeed jump onto the facets and continue their motion there, as was already shown in Fig. 3.95. An Arrhenius plot in Fig. 3.96 for their jump estimates showed that transitions along the facets could occur with an energy slightly higher than for single transitions along the troughs. Here it should also be noted that reconstruction of the (110) plane increased the activation energy for diffusion by 0.09 eV compared with the ordinary (110) surface studied by Kellogg [121].

4.12.2

Theoretical studies Observations by Bassett and Webber [120] of cross-channel movement triggered further calculational efforts. The second of these was done by Halicioglu and Pound [130,131] using Lennard-Jones potentials. For diffusion of Pt by exchange on Pt (110) they found a barrier of 1.07 eV. For Ir and Au they obtained 1.01 and 1.02 eV respectively, the former for two-dimensional exchange, the latter for in-channel jumps. They also looked at movement on the Pt(311) plane and obtained barrier heights of 0.656 eV, 0.734eV, and 0.486 eV for Pt, Ir, and Au atoms respectively. In 1980, Flahive and Graham [20] used Morse potentials to calculate activation energies, shown in Fig. 4.9. For self-diffusion on Pt(110) they got a value around 0.6 eV, for Pt(311) ~0.5 eV, and for Pt(331) ~0.7 eV. Liu et al. [2] were quite early in making estimates for activation energies of selfdiffusion on Pt(110) with EAM potentials. For in-channel motion evaluated with AFW interactions they obtained a low activation energy of 0.25 eV and a prefactor of 4 × 10−4 cm2/sec; in the VC approximation the results were 0.53 eV and 1.4 × 10−3 cm2/ sec. Cross-channel motion by atom exchange occurred over higher barriers, 0.43 eVand a prefactor of 7 × 10−3 cm2/sec using AFW potentials, and 0.68 eV with 1.5 × 10−2 cm2/sec for the prefactor calculated with the VC potentials. On the (311) plane the diffusion characteristics for in-channel motion were a barrier of 0.43 eV with a prefactor 8 × 10−4 cm2/sec (AFW), and 0.63 eV and 2.8 × 10−3 cm2/sec (VC). For movement in

234

Diffusion on one-dimensional surfaces

the channels of the (331) plane, the AFW approximation gave a barrier of 0.28 eV with a prefactor of 6 × 10−4 cm2/sec, and with VC potentials 0.54 eV and 8.5 × 10−4 cm2/sec. Liu et al. [2] also looked at diffusion with Leonard-Jones potentials, which brought much higher values. For in-channel movement on Pt(110) a barrier of 1.26 eV with a prefactor of 6 × 10−3 cm2/sec and for cross-channel motion 1.34 eV and a prefactor 3.4 × 10−2 cm2/sec. On the Pt(311) plane the barrier was 0.65 eV with a prefactor of 4 × 10−3 cm2/sec, on Pt(331) 1.33 eV and 5.5 × 10−3 cm2/sec. From L-J potentials only the value for Pt(311) was close to results obtained from many-body potentials or experiments, showing the importance of many-body interactions for diffusion studies. Estimates of platinum diffusion on Pt(110) were later made by Villarba and Jónsson [132], again with EAM potentials. For in-channel motion the barrier was estimated as 0.46 eV, and slightly higher, 0.58 eV, for cross-channel exchange. On the (311) plane the activation energy for diffusion was 0.57 eV, and along the (331) plane 0.53 eV. With Morse potentials they obtained a barrier of 0.53 eV along the channels of the (110) plane and 0.63 eVacross them. For diffusion on the (311) plane the Morse potential estimate for the barrier was 0.49 eVand 0.71 eVon the (331) plane. Prefactors were not investigated in this study. Relying on EMT interactions, Stoltze [27] calculated a self-diffusion barrier of 0.420 eV for in-channel jumps, and a much higher value of 0.809 eV for cross-channel diffusion by atom exchange. Cross-channel jumps were a bit more energetic, with a barrier of 0.945 eV, but again without any prefactors. Lorensen et al. [129] did density-functional estimates of the activation energy for diffusion on the reconstructed Pt(110)-(1 × 2). For diffusion along the bottom of the channel they obtained a barrier of 0.94 eV. When the adatom jumped onto the sidewalls, as illustrated in Fig. 3.95, the barrier was close to the same value, so that both paths were possible, as shown in Fig. 4.51. Feibelman [133] in the year 2000 made estimates of diffusion on Pt(110)-(1 × 2) using the VASP ab initio simulation [9–11]. For adatom diffusion via a metastable walk he arrived at a barrier of 0.93 eV, slightly higher than the value of 0.89 eV determined in experiments by Linderoth et al. [125,126]. Calculations with Lennard-Jones potentials for self-diffusion on Pt(110) were carried out by Agrawal et al. [7]. For platinum atom jumps along the channels, they derived a very high activation energy of 1.21 eVand a prefactor of 3.8 × 10−3 cm2/sec; for jumps across the channels they obtained a barrier of 2.08 eV, with a prefactor of 7.1 × 10−3 cm2/sec. These finding are not surprising at all, since it was already shown by Liu et al. [2] 11 years earlier that L-J potentials are not suitable for describing diffusion on this surface. Ndongmouo and Hontinfinde [32] worked with the more reliable RGL interactions and on the reconstructed (110)-(1 × 2) plane found a barrier to self-diffusion of 0.60 eV, and 0.54 eV on the (1 × 1) surface. For cross-channel atom exchange on the (1 × 1) Pt (110) plane the barrier proved to be 0.78 eV, and for jumps to a neighboring channel 1.38 eV, the former in good agreement with the experiments. Different diffusion barriers were obtained from molecular dynamics simulations using RGL potentials by Kim et al. [34]. For in-channel self-diffusion they found an energy of 0.490 eV was required, for cross-channel exchange the barrier was 0.779, and for crosschannel jumping 1.247 eV. The value for cross-channel motion by exchange seems to be quite high compared with the in-channel barrier. Also determined were the barriers for

4.13 Gold: Au(110), (311), (331)

235

in-channel hetero-motion on Pt(110) – 0.463 eV for Ni, 0.363 eV for Cu, 0.402 eV for Pd, 0.376 eV for Ag, and 0.386 eV for Au. The value for gold is much too low compared with the experimental findings of Bassett and Webber [120], the value for Pd is lower than the experimental results of Kellogg [124]. Not too much experimental data about self-diffusion on Pt(110) is available, but the two existing reports are not that far apart and it is clear that both mechanisms, hopping and exchange, participate on this surface. The various theoretical efforts are quite different, and even the RGL as well as EMT estimates differ significantly and favor jumping over exchange, which is not seen in the experiments. One thing is clear – manybody potentials must be used in studying platinum surfaces.

4.13

Gold: Au(110), (311), (331) Hardly any experimental work has been done to measure atomic diffusivities on Au(110). Günther et al. [134], in 1997, carried out STM studies of atom hopping on Au(110). From observations at two temperatures and a coverage of 0.2 ML, they found that on the (1 × 2) reconstructed surface the activation energy for self-diffusion ranged from 0.40 eV to 0.44 eV, with a frequency prefactor of 1012– 1013 sec−1. On the (1 × 1) surface, prepared from the reconstructed plane after atoms had locally filled in the wide troughs, the minimum barrier height was lowered to 0.38 eV. Hitzke et al. [135] in the same year looked briefly at diffusion of Ni on Au(110)-(1 × 2) using the STM. They examined a surface covered with a 0.22 ML film of Ni at 180 K, and deduced a barrier of 0.52 ± 0.1 eV along , on the assumption of a frequency prefactor of 1013 sec−1. However, in both experimental studies movements are likely to be influenced by adatom interactions. There have, however, been a number of attempts to calculate the diffusion characteristics. In 1980 Flahive and Graham [20] did calculation using Morse potentials with results shown in Fig. 4.9. Early estimates of self-diffusion on Au(110) were made by Roelofs et al. [136] in 1991, who did Monte Carlo simulations with interactions similar to Morse potentials. From an Arrhenius plot of in-channel motion they found an activation energy of only 0.140 eV; for cross-channel movement by atom exchange the barrier was 0.305 eV. Four years later Roelofs et al. [137] continued estimates of diffusion, this time based on EAM interactions. The activation energy for in-channel motion was given as 0.27 eV. For the cross-channel exchange process the barrier was 0.35 eV, with a frequency prefactor of ~0.76 × 1012 sec−1. Liu et al. [2] in 1991 used EAM interactions and transition state theory to reach an in-channel barrier of 0.25 eV and a prefactor of 6 × 10−4 cm2/sec with AFW parameters; for VC potentials the values were 0.34 eV and 1.6 × 10−3 cm2/sec. In cross-channel motion the activation energies were 0.40 eV with a prefactor of 2.5 × 10−2 cm2/sec (AFW) and 0.42 eV and 1.3 × 10−2 cm2/sec (VC). They also did calculations for Au(311) and Au(331) where movement is likely to be one-dimensional. For the (311) plane using AFW potentials they obtained 0.35 eV and a prefactor of 8 × 10−4 cm2/sec, from VC potentials the barrier to diffusion was 0.42 eV with a prefactor of 3.6 × 10−3 cm2/sec. For the (331) plane, the barriers were 0.26 eV and the prefactor 6 × 10−4 cm2/sec (AFW) and 0.34 eV and 9.6 × 10−4 cm2/sec (VC). These are the only determinations for the two planes, making comparison impossible.

236

Diffusion on one-dimensional surfaces

With effective medium theory potentials, Stoltze [27] calculated a self-diffusion barrier of 0.268 eV for in-channel jumps on Au(110); for cross-channel exchange the activation energy proved to be higher, 0.554 eV, and the jump across the channel was only slightly more energetic at 0.67 eV. Montalenti and Ferrando [127] in 1998 made an important discovery in molecular dynamics simulations of self-diffusion on the reconstructed Au(110)-(1 × 2) surface. Representing interactions in the RGL approximation, they found that, as shown in Fig. 3.95, the atom could jump from the bottom of the channel to the (111)-like sidewalls, where diffusion continued until the atom again descended. From Arrhenius plots of the diffusion rate they found a barrier of 0.29 ± 0.01 eV for single jumps, and 0.43 ± 0.03 eV for double jumps, both along the bottom of the troughs. In the metastable excursions on the wall, they discovered a comparable activation energy of 0.37 ± 0.02 eV for single jumps, and 0.43 ± 0.05 eV for double jumps. Jumps along the channel bottoms and along the sides were therefore both likely to contribute to diffusion. They also looked at the static energy and for single jumps they obtained a barrier of 0.31 eV. Climbing a wall required 0.4 eV, but the opposite motion needed only 0.06 eV; movement on the (111) wall required overcoming a barrier of 0.10 eV. Long jumps in diffusion were explored by Montalenti and Ferrando [45,128] in molecular dynamics simulations using RGL potentials. For in-channel motion of Au on Au(110) they found a barrier of 0.28 eV, and 0.46 eV for cross-channel motion with atom exchange; cross-channel jumps cost much more, with a barrier of 0.7 eV. For diffusion by in-channel jumps on the reconstructed Au(110)-(1 × 2) surface, the barrier was higher, 0.31 eV; cross-channel exchange cost 0.66 eV, cross-channel jumps were comparable to exchange at 0.63 eV. The effective barrier for metastable walks was 0.44 eV. An Arrhenius plot, in Fig. 4.52, showed a nice straight line, and in a detailed examination of jump statistics longer jumps amounted to less than 1% of the total at 450 K. In 2002 Agrawal et al. [7] used MC variational transition state theory and LennardJones potentials to calculate the different barriers on Au(110). They found a barrier of 0.86 eV for in-channel movement, much higher than the usual estimate, and a prefactor –4.5 –5.0

In R

–5.5 –6.0 –6.5 –7.0 –7.5 –8.0 1.6

Fig. 4.52

ED = 0.288 ± 0.005 eV 1.8

2.0 2.2 1000/T (K–1)

2.4

2.6

Dependence of Au atom diffusivity on Au(110)-(1 × 2) surface upon reciprocal temperature, derived from molecular dynamics simulations with RGL potentials. Static barrier 0.31 eV, from Arrhenius plot 0.288 ± 0.005 eV (after Montalenti and Ferrando [45]).

4.13 Gold: Au(110), (311), (331)

237

6.3 × 10−3 cm2/sec; for cross-channel movement they calculated a barrier of 1.49 eV with a prefactor of 2.8 × 10−2 cm2/sec. Ndongmouo and Hontinfinde [32,138] used RGL potentials to calculate a barrier for self-diffusion jumps on Au(110)-(1 × 2) of 0.31 eV. For Au (1 × 1) they found a smaller barrier of height 0.28 eV. For cross-channel diffusion by exchange the barrier was 0.46 eV, and for cross-channel jumps it proved to be higher, 0.69 eV. The same RGL potentials were used by Kim et al. [34] in molecular dynamics estimates of diffusion on Au(110). For in-channel motion the barrier was 0.274 eV, for exchange 0.468 eV, and 0.700 eV for cross-channel jumps, close to the previous estimates. In hetero-diffusion of Ni the barrier proved to be 0.313 eV, for Cu 0.243 eV, for Pd 0.281 eV, and Pt 0.393 eV. Molecular dynamics simulations of nickel adatom diffusion on Au(110)-(1 × 2) were done by Fan and Gong [139] using embedded atom potentials from Johnson [140,141]. Diffusion can of course proceed in different ways, along the troughs of the channels making up this plane, or by moving up to the channel facets and diffusing there. This is demonstrated by the atom trajectories shown in Fig. 4.53. For in-trough motion the activation energy was 0.21 eV; for motion to a position on the sidewalls the barrier was comparable, 0.20 eV. However diffusion in the opposite direction, from the wall to the bottom, required only 0.04 eV. Diffusion on the wall occurred over a small barrier of 0.05 eV between points C and B as well as B and C in Fig. 4.54, and 0.08 eV for C to F. Diffusion on the sidewalls took place with a much smaller activation energy, and can be expected to play a significant role. Calculated diffusion characteristics for Au(110) are in reasonable agreement with each other, but more experimental information would certainly be desirable, as the available

Fig. 4.53

Ni atom trajectories on Au(110)-(1 × 2) surface, obtained by molecular dynamics simulations with Johnson potentials [140,141]. (a) Single in-channel jump. (b) Jumps on channel walls. (c) Diffusion over local sidewall minima. (d) Diffusion from one channel to the next (after Fan and Gong [139]).

238

Diffusion on one-dimensional surfaces

Fig. 4.54

Potential energy for Ni atom on Au(110)-(1 × 2) surface [139]. Filled circles show atoms at top of channel, open circles atoms at channel bottom. Minimum energy locations A, B, C; saddle points E, D (after Fan and Gong [139]).

experimental data differ from calculated values. Also information about hetero-diffusion is quite scarce.

4.14

Lead: Pb(110) Quasi-elastic helium scattering studies were carried out by Frenken et al. [142] on the Pb (110) surface at close to the melting point. The diffusivity was deduced from the energy distribution of the helium atoms scattered from the surface. The activation energy for diffusion was derived as 0.65 ± 0.2 eV, with a prefactor Do = 26 cm2/sec, and the diffusivity along and across the surface channels was roughly the same. After additional studies at temperatures from ~300 K to ~560 K, Frenken et al. [143] came up with a diffusion barrier of 1.0 ± 0.3 eV and a prefactor of 6.2 × 104 cm2/sec for cross-channel motion. For motion along the channels they tentatively estimated an activation energy of 1.0 eV and a prefactor of 1.8 × 105 cm2/sec. These numbers all look to be too large by comparison with other materials, and it does not appear that single atom motion was being measured. The activation energy seems to be not just for atom motion, but may also have a contribution from the energy for forming vacancies. Calculations of the self-diffusion barrier on Pb(110) have been made by Karimi et al. [144] using EAM interactions. For diffusion along the channels they arrived at a barrier of only 0.215 eV; for cross-channel motion by exchange the barrier was higher, 0.454 eV. Adequate experiments to test the validity of these estimates are not available. It is clear that more experimental as well as calculational work needs to be done to achieve insight into the characteristics of self-diffusion on Pb(110).

4.15

Tables for 1D Diffusion To provide perspective as well as a ready reference and also to allow quick comparisons, much of the diffusion data is summarized here in tabular form. To ease access, the material is subdivided according to the substrate plane for which results are presented.

Table 4.1 Fcc(110) Experiment System Al/Al(110)

Mechanism ED (eV)

Do (cm2/sec)

k Jump

†

?Ex

0.43

0.43

†

10

10

k Jump

0.23 ± 0.04 0.30 ± 0.06

Do (cm2/sec)

0.26 0.204 0.33 0.67 0.38

1.8 × 10−3

0.30 0.15 0.285 0.62 0.50 1.19 0.83

6.0 ×10−2 2.4 × 10−2

−3

−3

? Jump Ni/Ni(110)

Theory ED (eV)

1 × 10−9 101

0.45 0.536 ± 0.040 0.60 0.44 0.39 0.50 0.53 0.44 0.64 0.41 0.407 0.243 0.18 0.39

3.5 × 10−3

6.9 × 10−3

†

10−3

2.3 × 10−3 4.0 × 10−3 3.3 × 10−3 3.8 × 10−3 3.6 × 10−3 4.5 × 10−3

ν0 (sec−1)

Method

Ref.

FIM EAM AFW EMT DFT L-J VASP FIM EAM AFW EAM VC EMT DFT VASP L-J VASP FIM‡ FIM‡‡ Refit He Scat STM EAM AFW EAM VC EAM FBD EAM VC EAM OJ EAM ATVF EAM EMT CEM¥ CEM EAM5

[1] [2] [5] [6] [7] [8] [1] [2] [2] [5] [6] [8] [7] [8] [12] [12] [2] [15,16] [17] [2] [2] [23] [23] [23] [23] [22] [27] [28] [28] [29]

Table 4.1 (cont.) Experiment System

Mechanism ED (eV)

? Ex

0.32 ± 0.05 0.26 ± 0.06

Theory 2

Do (cm /sec)

10−7 10−1

ED (eV) 0.39 0.86 0.32 0.301

0.45 0.424 ± 0.040

? Jump

Cu/Ni(110) Pd/Ni(110) Ag/Ni(110) Pt/Ni(110) Au/Ni(110) Cu/Cu(110)

k Jump k Jump k Jump ?Ex k Jump k Jump k Jump

0.28

†

10−3

0.49 0.42 0.564 0.392 0.35 0.42 0.38 0.301 1.30 1.67 1.01 1.53 1.157 1.68 1.33 1.026 0.334 0.426 0.334 0.308 0.440 0.23

ν0 (sec−1)

Do (cm2/sec) −3

(1.4 ± 0.1) × 10 2.5 × 10−3

†

10−3

3.7 × 10−3 2.8 × 10−2

6.5 × 10−3 7.5 × 10−3 6.5 × 10−3 7.7 × 10−3 4.0 × 10−2

8 × 10−4

Method

Ref.

EAM L-J RGL RGL FIM‡ FIM‡‡ Refit He Scat EAM AFW EAM VC EMT CEM¥ CEM EAM5 RGL RGL EAM FBD EAM VC EAM OJ EAM ATVF EMT L-J RGL RGL RGL RGL RGL FIM RGL RGL EAM AFW

[31] [7] [32] [34] [12] [12] [2] [15,16] [2] [2] [27] [28] [28] [29] [32] [34] [23] [23] [23] [23] [27] [7] [32] [34] [34] [34] [34] [13] [34] [34] [2]

k Jumpÿ α ? Move ? Ex

0.28 0.18 0.26 0.08 0.292 0.24 0.23 0.47 0.25 0.251 0.25 0.76 0.23 0.23 0.241 0.23 0.26 0.16 ± 0.01 0.213 ± 0.007 0.84 ± 0.04 0.30 0.31 0.27 0.485 0.09 0.419 0.30 0.29 0.63 0.30 0.284 0.27 0.34 0.323 0.30

4.4 × 10−3

(1.1 ± 0.1) × 10−3 7.5 × 10−3 6.29 × 10−4 4.0 × 10−3 2 × 1012 9 × 1012 3.2 × 10−2 2.7 × 10−2

EAM VC EMT CEM¥ CEM59 EMT EAM AFW RGL RGL RGL RGL RGL L-J RGL EAM RGL EAM MS RGL RGL He Scat EAM AFW EAM VC EMT CEM¥ CEM59 EMT EAM AFW RGL RGL RGL RGL RGL EAM RGL MS

[2] [39] [28] [28] [27] [40] [44] [42] [43] [46] [31] [7] [32] [48] [34] [50] [51] [45] [45] [37] [2] [2] [39] [28] [28] [27] [40] [44] [42] [43] [46] [32] [49] [34] [51]

Table 4.1 (cont.) Experiment System

Mechanism ED (eV)

Theory 2

Do (cm /sec)

? Jump

Co/Cu(110)

k Jump ? Ex

Ni/Cu(110) Pd/Cu(110) Ag/Cu(110) Pt/Cu(110) Au/Cu(110)

Pb/Cu(110)

Rh/Rh(110)

? Jump k Jump k Jump k Jump k Jump k Jump ? Ex k Jump α ? Ex k Jump

0.60 ± 0.03

? Ex

0.826 1.15 1.31 1.07 1.146 1.020 1.146 0.29 0.35 0.30 0.32 >1.0 0.244 0.318 0.254 0.376 0.19 ± 0.03 0.333 0.23 ± 0.04 0.212 0.14 0.281 1.24 0.48 1.20 0.78 0.91 2.06 1.80

? Ex ? Jump Pt/Rh(110)

3 × 10−1

ED (eV)

~0.65

Do (cm2/sec)

2.6 × 10−2 9.97 × 10−4 7.4 × 10−3

8 × 10−3 4.4 × 10−3 1.0 × 10−2

ν0 (sec−1)

Method

Ref.

EMT EAM L-J RGL EAM RGL EAM MS VASP MS VASP MS VASP RGL RGL RGL RGL RGL RGL RGL RGL-Arrh RGL FIM L-J Morse L-J RGL RGL L-J RGL FIM

[27] [40] [7] [32] [48] [34] [50] [51] [51] [51] [51] [51] [34] [34] [34] [34] [42] [34] [42] [46] [46] [46] [60] [2] [2] [7] [32] [32] [7] [32] [61]

Pd/Pd(110)

k Jump

0.28 0.45 0.28 0.30 0.30 0.28 0.366 0.72 0.39 0.38 0.42 0.42 0.34 0.33 0.38 0.599 0.54 0.551 0.776 1.56 1.35 0.99 0.965 0.369

? Ex

? Jump

Ni/Pd(110) Cu/Pd(110)

k Jump k Jump

0.51 ± 0.05 0.32 0.3 0.315

?Move ? Ex

0.75 ± 0.07 0.57 0.45

Ag/Pd(110) Pt/Pd(110) Au/Pd(110) Ag/Ag(110)

? Jump k Jump k Jump k Jump k Jump

1.25 0.333 0.520 0.392 0.279

−4

4 × 10 3.5 × 10−3

1.8 × 10−3

3.3 × 10−2 2.4 × 10−2

9.2 × 10−3

EAM Morse EAM AFW EAM VC CEM CEM¥ EMT L-J RGL RGL EAM EAM AFW EAM VC CEM CEM¥ EMT RGL RGL EMT EAM L-J RGL RGL RGL STM EAM STM RGL STM EAM STM EAM RGL RGL RGL STM

[63] [20] [2] [2] [28] [28] [27] [7] [32] [34] [63] [2] [2] [28] [28] [27] [32] [34] [27] [63] [7] [32] [34] [34] [62] [62] [65] [34] [62] [62] [65] [62] [34] [34] [34] [66]

Table 4.1 (cont.) Experiment System

Mechanism ED (eV)

Theory 2

Do (cm /sec)

ED (eV)

0.30 ± 0.03

k Jumpa k Jumpb k Jumpc ? Ex

? Move ? Jump

Ag/Ag(110)-(2 × 2)

Ni/Ag(110) Cu/Ag(110) Pd/Ag(110)

k Jump ? Ex ? Jump k Jump k Jump k Jump

Do (cm2/sec)

0.32 0.25 0.248 0.26 0.291 0.28 0.59 0.277

1.0 × 10−3 2.7 × 10−3

0.42 0.31 0.328 0.34 0.561 0.38 0.388

4.0 × 10−2 2.5 × 10−2

0.19 ± 0.05 0.26 ± 0.03 0.29 ± 0.02 0.394

2.6 × 10−3

~0.35 0.639 0.82 1.10 0.818 0.266 ~0.4 0.757 0.268 0.249 0.337

6.9 × 10−3

ν0 (sec−1)

Method

Ref.

STM EAM AFW EAM VC CEM¥ CEM EMT RGL L-J RGL He Scat He Scat He Scat STM EAM AFW EAM VC CEM¥ CEM EMT RGL RGL STM EMT RGL L-J RGL DFT DFT DFT RGL RGL RGL

[67] [2] [2] [28] [28] [27] [75] [7] [34] [68] [69,70] [71] [66] [2] [2] [28] [28] [27] [75] [34] [67] [27] [75] [7] [34] [78] [78] [78] [34] [34] [34]

Pt/Ag(110) Au/Ag(110) Ir/Ir(110)

k Jump k Jump ? Ex

k Jump

0.74 ± 0.09 0.71 ± 0.02

0.80 ± 0.04

±1

4.0 × 10−3 ± 0.8

Ir/Ir(110)-(1 × 2)

k Jump

0.86 ± 0.02

1.2 × 10−3 ± 1

Pt/Pt(110)

k Jump

0.84 ± 0.1 0.72 ± 0.07

8 × 10−3 6 × 10−4

0.78 ± 0.1

0.492 0.360

1.4(× 16 ) × 10 6 × 10−2 ± 1.8

?Jump

?Ex

−6

1 × 10−3

0.92 0.75 0.81 1.60 ~0.75 0.70 0.83 0.70 1.48 1.15 2.58 2.5 2.47 2.21

4.7 × 10−3

1.1 × 10−2

1.22

0.64 0.6 0.25 0.53 1.26 0.46 0.53 0.42 1.21 0.54 0.49

4 × 10−4 1.4 × 10−3 6 × 10−3

3.8 × 10−3

RGL RGL FIM FIM EAM RGL EAM RGL FIM Morse EAM RGL EAM L-J RGL RGL EAM L-J RGL FIM RGL FIM FIM Morse Morse EAM AFW EAM VC L-J EAM Morse EMT L-J RGL RGL FIM

[34] [34] [114] [41] [118] [118] [119] [32] [41] [20] [118] [118] [119] [7] [32] [118] [119] [7] [32] [41] [32] [120] [121] [120] [20] [2] [2] [2] [132] [132] [27] [7] [32] [34] [120]

Table 4.1 (cont.) Experiment System

Theory 2

Mechanism ED (eV)

Do (cm /sec)

0.69 ± 0.07

3 × 10

0.43 0.68 1.34 1.07 0.58 0.63 0.809 0.78 0.779 1.97 0.945 2.08 1.38 1.247

? Jump

Pt/Pt(110) − (1 × 2)

k Jump

ED (eV)

Do (cm2/sec)

ν0 (sec−1)

Method

Ref.

1010.7 ± 0.2

FIM EAM AFW EAM VC L-J L-J EAM Morse EMT RGL RGL Morse EMT L-J RGL RGL STM* DFT RGL STM* VASP DFT DFT DFT DFT RGL RGL FIM RGL RGL FIM Morse

[121] [2] [2] [2] [130] [132] [132] [27] [32] [34] [120] [27] [7] [32] [34] [125] [129] [32] [125] [133] [129] [129] [129] [129] [34] [34] [124] [34] [34] [120] [120]

−4

0.81 ± 0.01

7 × 10−3 1.5 × 10−2 3.4 × 10−2

7.1 × 10−3

0.94 0.60 MW

Ni/Pt(110) Cu/Pt(110) Pd/Pt(110) Ag/Pt(110) Ir/Pt(110)

α Jump β Jump γ Jump k Jump k Jump k Jump k Jump k Jump k Jump

1010.9 ± 0.9

0.89 ± 0.06

3.5 × 10−3±1.2

0.58 ± 0.05

0.80 ± 0.15

£

†

10

−5

0.93 ~0.94 0.92 0.95 0.98 0.463 0.363 0.402 0.376 0.66

? Ex Au/Pt(110)

Au/Au(110)

k Jump

0.80 ± 0.15£ 0.63 ± 0.15

£

†

10

−7

1.01 0.57 1.02 0.386 0.14 0.25 0.34 0.268 0.27

k Jump

6 × 10−4 1.6 × 10−3

0.38 0.28 0.86 0.28 0.274 0.305 0.40 0.42 0.554 0.35 0.46 0.46 0.468 0.67 0.7 1.49 0.69 0.70

? Ex

? Jump

Au/Au(110) − (1 × 2) k Jump k α Jump β Jump Meta α

6.3 × 10−3

2.5 × 10−2 1.3 × 10−2 ~0.76 × 1012

2.8 × 10−2 1012–1013

0.40–0.44 0.31 0.29 ± 0.01 0.31 0.43 ± 0.03 0.37 ± 0.02

FIM L-J FIM Morse L-J RGL Morse EAM AFW EAM VC EMT EAM STM RGL L-J RGL RGL Morse EAM AFW EAM VC EMT EAM RGL RGL RGL EMT RGL L-J RGL RGL STM RGL RGL RGLS RGL RGL

[120] [130] [120] [120] [130] [34] [136] [2] [2] [27] [137,145] [134] [45,128] [7] [32] [34] [136] [2] [2] [27] [137,145] [45,128] [32] [34] [27] [45,128] [7] [32] [34] [134] [32] [127] [45,128] [127] [127]

Table 4.1 (cont.) Experiment System

Mechanism ED (eV)

Ni/Au(110) Ni/Au(110)-(1 ×2)

Meta β MW ? Ex ? Jump k Jump k Jump

Cu/Au(110) Pd/Au(110) Pt/Au(110) Pb/Pb(110)

Theory 2

Do (cm /sec)

ED (eV) 0.43 ± 0.05 0.44 0.66 0.63 0.313

0.52 ± 0.1 0.21 0.29 0.243 0.281 0.393

M-Jump k Jump k Jump k Jump 0.65 ± 0.2

26

1.0 1.0 ± 0.3d

1.8 × 105 6.2 × 104

0.215 k Jump ? Exd ? Ex ‡

d

0.454 ‡‡

Do (cm2/sec)

ν0 (sec−1)

†

1013

Method

Ref.

RGL RGL RGL RGL RGL STM EAM EAM RGL RGL RGL He Scat EAM He Scat He Scat EAM

[127] [45,128] [45,128] [45,128] [34] [135] [139] [139] [34] [34] [34] [142] [144] [143] [143] [144]

* direct measurement; after thermal treatment; after hydrogen treatment; CEM¥ – MD/MC CEM; a Temperature range 600–750 K; b Temperature range 650–775 K; c Temperature range 600–800 K; d Temperature range 300–560 K; † assumed value; £ assumed kT‘2 /2h; S static estimation; ÿ above 400 K.

Table 4.2 Fcc(311) Experiment System Al/Al(311)

Ni/Ni(311)

Mechanism k Jump k Jump

Pt/Ni(311) Cu/Cu(311)

? Ex ? Jump k Jump k Jump

Rh/Rh(311)

? Ex ? Jump k Jump

Pt/Rh(311) Pd/Pd(311)

k Jump k Jump

Ag/Ag(311)

k Jump

ED (eV) 0.48

0.30 ± 0.03

0.38

0.54 ± 0.04

0.44 ± 0.05

Theory 2

Do(cm /sec) †

10

ED (eV)

Do(cm2/sec)

−3

1.9 × 10−6

†

10−3

2 × 10−3 (1.6 ± 0.5) × 10−4

0.20 0.24 0.37 0.34 0.38 0.351 1.302 1.373

2.0 × 10−3 6.7 × 10−3 †

10−3 1.4 × 10−3 4.4 × 10−3

0.26 0.28 0.232 0.889 0.822

1.2 × 10−3 3.1 × 10−3

0.64 0.44

5.4 × 10−3

~0.35 0.37 0.41 0.26

1.3 × 10−3 3.1 × 10−3 3.0 × 10−3

ν0 (sec−1)

Method

Ref.

FIM EAM AFW EAM VC FIM Refit EAM AFW EAM VC EMT EMT EMT FIM EAM AFW EAM VC EMT EMT EMT FIM L-J Morse FIM Morse EAM AFW EAM VC EAM VC

[1] [2] [2] [12] [2] [2] [2] [36] [36] [36] [14] [2] [2] [36] [36] [36] [60] [2] [2] [61] [20] [2] [2] [2]

Table 4.2 (cont.) Experiment System

Mechanism

ED (eV)

Theory

Do(cm2/sec)

Ir/Ir(311)

? Ex ? Jump k Jump

0.72 ± 0.02

1 × 10−3 ± 0.4

Pt/Pt(311)

k Jump

0.69 ± 0.2£

†

0.60 ± 0.03

1.9 × 10−4

ED (eV)

Do(cm2/sec)

0.26 0.22 0.727 0.704

1.0 × 10−3

~0.6

10−6

0.50 ~0.50 0.656 0.43 0.63 0.65 0.57 0.49

Ir/Pt(311)

Au/Pt(311)

Au/Au(311)

£

k Jump k Jump k Jump

assumed kT‘2 /2h; † assumed value.

8 × 10−4 2.8 × 10−3 4 × 10−3

0.74 ± 0.15£

0.56 ± 0.15

3 × 10

−1

0.54 0.734 0.41 0.486 0.35 0.42

8 × 10−4 3.6 × 10−3

ν0 (sec−1)

Method

Ref.

EAM AFW EMT EMT EMT FIM Morse FIM Morse FIM Morse L-J EAM AFW EAM VC L-J EAM Morse FIM Morse Morse FIM Morse Morse EAM AFW EAM VC

[2] [36] [36] [36] [117] [20] [120] [120] [122] [20] [130] [2] [2] [2] [132] [132] [120] [120] [130] [120] [120] [130] [2] [2]

251

4.15 Tables for 1D Diffusion

Table 4.3 Fcc(331) Experiment

Theory

System

Mechanism ED (eV)

Do (cm2/sec) ED (eV) Do (cm2/sec) ν0 (sec−1) Method

Al/Al(331)

k Jump

†

Ni/Ni(331)

k Jump

0.46

10−3

0.45 ± 0.03 2.3 × 10−3

Cu/Cu(331) k Jump Rh/Rh(331) k Jump Pt/Rh(331) k Jump Pd/Pd(331) k Jump

0.64 ± 0.04 1 × 10−2

0.72 ± 0.02

†

10−3

Ag/Ag(331) k Jump Ir/Ir(331)

k Jump

0.91 ± 0.03 1 × 10−1.9 ± 0.5

Pt/Pt(331)

k Jump

0.84 ± 0.1

Ir/Pt(331) k Jump Au/Pt(331) k Jump Au/Au(331) k Jump †

assumed value.

4 × 10−4

0.27 0.24 0.45 0.45 0.46 0.28 0.33

2.0 × 10−3 5.4 × 10−3 †

10−3 1.4 × 10−3 4.2 × 10−3 1.0 × 10−3 3.0 × 10−3

1.31 0.62

7.4 × 10−3

0.33 0.37 0.55 0.34 0.29

8 × 10−4 2 × 10−3 1.0 × 10−3 2.0 × 10−3

0.95 ~0.7 0.77 0.28 0.54 1.33 0.53 0.71 0.81 0.68 0.26 0.34

6 × 10−4 8.5 × 10−4 5.5 × 10−3

6 × 10−4 9.6 × 10−4

FIM EAM AFW AFW VC FIM Recal. EAM AFW EAM VC EAM AFW EAM VC FIM L-J Morse FIM EAM AFW EAM VC Morse EAM AFW EAM VC FIM Morse FIM Morse Morse EAM AFW EAM VC L-J EAM Morse Morse Morse EAM AFW EAM VC

Ref. [1] [2] [2] [12] [2] [2] [2] [2] [2] [60] [2] [2] [61] [2] [2] [20] [2] [2] [117] [20] [120] [20] [120] [2] [2] [2] [132] [132] [120] [120] [2] [2]

Table 4.4 Bcc (211) Experiment

Theory 2

System

Mecha-nism

ED (eV)

Do (cm /sec)

Li/Mo(211) Cu/Mo(211) Sr/Mo(211) Dy/Mo(211) W/W(211)

kJump kJump k Jump k Jump kJump

0.6 0.45 0.9 1.4 0.85 ± 0.05 0.82 ± 0.02 0.81 ± 0.02 0.81 ± 0.02 0.56 0.73 ± 0.02

2.5 1.2 × 10−3 1.8 102 19.0(× 7.3±1) × 10−3 7.7(× 1.9±1) × 10−3 2.5(× 3.3±1) × 10−3 3.41(× 2.4±1) × 10−3 4.8 × 10−7 (3–2) × 10−5

Sr/W(211) Ni/W(211)

k Jumpe ? Move ? Movee ? Ex α Jump β Jump βR Jump k Move k Move ? Move k Move k Jump

Mo/W(211)

k Jump

Rh/W(211) Pd/W(211)

k Jump k Jump

Ta/W(211) Re/W(211)

k Jump k Jump

Ir/W(211)

k Jump

Li/W(211) K/W(211)

Au/W(211)

0.95 ± 0.02 0.29 1.04 0.84 ± 0.06 1.44 ± 0.13 1.03 ± 0.06 0.49f 0.46 ± 0.08 0:76þ0:10
0:08 0.87 0.46 ± 0.01 0.57 0.71 ± 0.01 0.536 ± 0.001 0.32 ± 0.01 0.32 ± 0.02 0.48 0.86 ± 0.03 0.83 ± 0.01 0.53 ± 0.05 0.67 ± 0.01 0.19¢

5 × 10−5 4.2 × 10−9 5 × 10−4

ED (eV)

Do (cm2/sec)

0.78 0.432 0.28 0.887 ± 0.224 0.952 ± 0.182 0.822 0.79

2.8 × 10−3 1.6 × 10−3 1.41 × 10−3

2.00

9.4 × 10−4

2.0 × 10−3 ~ 0.3 ~ 0.3 0.4 1.9(× 2.7±1) × 10−3 9.3 × 10−7 2.0(× 1.6±1) × 10−3 3.3(× 1.5±1) × 10−3 3.9(× 2.4±1) × 10−4 5.1 × 10−4.0 ± 0.8 0.9 × 10−7 2.2(× 2.8±1) × 10−3 0.73(× 1.6±1) × 10−3 5.0 × 10−7 0.61(× 1.5±1) × 10−3

** multi atom measurement; e High temperature; f coverage 0.4 ML; ¢ coverage 0.5 ML.

ν0 (sec−1)

2.2(× 11.28±1) × 1013 7.90(127.3±1) × 1021 1.4(× 10.3±1) × 1016

Method

Ref.

WF WF WF WF Rean FIM FIM FIM FEM FEM Morse Morse Morse L-J L-J MC L-J TST 4th Moment FEM FEM FEM 4th Moment FIM FIM FIM FEM SI SI WF FIM

[56] [59] [57] [58] [86] [86] [87,88] [89] [92] [93] [95] [96] [20] [97] [98] [98] [99] [93] [93] [93] [99] [89] [89] [89] [110] [109] [109] [111] [88]

FIM FIM FIM FIM FIM FIM** FIM FIM FIM FIM FEM

[100] [86] [86] [87,102] [103] [107] [105] [86] [106] [86] [108]

254

Diffusion on one-dimensional surfaces

Table 4.5 BCC(321) Experiment System

Mechanism ED (eV)

W/W(321)

k Jump

Re/W(321)

F->I I->I 2-D 2-D ? Ex F->I k Jump

Ta/W(321) k Jump Mo/W(321) k Jump

Theory

Do (cm2/sec) ED (eV) Do(cm2/sec) ν0 (sec−1) Method

0.81 ± 0.078 1 × 10−4 0.89

0.65 ~0.87 0.35 0.71 0.88 0.88 0.67 0.55

†

10−3 † 10−3 1.8 × 10−8 1.4 × 10−5 −3

0.5 × 10 4.8 × 10−4 1.9 × 10−5 1.2 × 10−7

0.91 0.13 0.73

1.36 × 10−3

0.23

3.4 × 10−4

FIM FIM Morse Morse 4th Moment FIM FIM FEM FEM 4th Moment FIM** FIM** FIM** FIM

Ref. [84] [112] [95] [20] [99] [85] [85] [92] [92] [99] [104] [107] [107] [100]

** multi atom measurement; † assumed value.

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260

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[125] T. R. Linderoth, S. Horch, E. Laegsgaard, I. Stensgaard, F. Besenbacher, Surface diffusion of Pt on Pt(110): Arrhenius behavior of long jumps, Phys. Rev. Lett. 78 (1997) 4978–4981. [126] T. R. Linderoth, S. Horch, E. Laegsgaard, I. Stensgaard, F. Besenbacher, Dynamics of Pt adatoms and dimers on Pt(110)-(1 × 2) observed directly by STM, Surf. Sci. 402–404 (1998) 308–312. [127] F. Montalenti, R. Ferrando, Competing mechanisms in adatom diffusion on a channeled surface: Jumps versus metastable walks, Phys. Rev. B 58 (1998) 3617–3620. [128] F. Montalenti, R. Ferrando, An MD study of adatom self-diffusion on Au(110) surfaces, Surf. Sci. 433–435 (1999) 445–448. [129] H. T. Lorensen, J. K. Nørskov, K. W. Jacobsen, Mechanism of self-diffusion on Pt(110), Phys. Rev. B 60 (1999) R5149–5152. [130] T. Halicioglu, G. M. Pound, A calculation of the diffusion energies for adatoms on surfaces of FCC metals, Thin Solid Films 57 (1979) 241–245. [131] T. Halicioglu, An atomistic calculation of two-dimensional diffusion of a Pt adatom on a Pt(110) surface, Surf. Sci. 79 (1979) L346–348. [132] M. Villarba, H. Jónsson, Diffusion mechanisms relevant to metal crystal growth: Pt/Pt(111), Surf. Sci. 317 (1994) 15–36. [133] P. J. Feibelman, Ordering of self-diffusion barrier energies on Pt(110)-(1 × 2), Phys. Rev. B 61 (2000) R2452–2455. [134] S. Günther, A. Hitzke, R. J. Behm, Low adatom mobility on the (1 × 2)-missing-row reconstructed Au(110) surface, Surf. Rev. Lett. 4 (1997) 1103. [135] A. Hitzke, M. B. Hugenschmidt, R. J. Behm, Low temperature Ni atom adsorption on the Au(110)-(1 × 2) surface, Surf. Sci. 389 (1997) 8–18. [136] L. D. Roelofs, J. I. Martin, R. Sheth, Competition between direct and concerted movements in surface diffusion with application to the Au(110) surface, Surf. Sci. 250 (1991) 17–26. [137] L. D. Roelofs, B. J. Greenblatt, N. Boothe, Kinetic prefactors for concerted-mode diffusion: a realistic calculation – Au/Au(110), Surf. Sci. 334 (1995) 248–256. [138] U. T. Ndongmouo, F. Hontinfinde, R. Ferrando, Numerical study of the stability of (111) and (331) microfacets on Au, Pt, and Ir(110) surfaces, Phys. Rev. B 72 (2005) 115412 1–8. [139] W. Fan, X. G. Gong, Simulation of Ni cluster diffusion on Au(110)-(1 × 2) surface, Appl. Surf. Sci. 219 (2003) 117–122. [140] R. A. Johnson, Analytical nearest-neighbor model for fcc metals, Phys. Rev. B 37 (1988) 3924–3931. [141] R. A. Johnson, Alloy models with the embedded atom method, Phys. Rev. B 39 (1989) 12554–12559. [142] J. W. M. Frenken, J. P. Toennies, C. Wöll, Self-diffusion at a melting surface observed by He scattering, Phys. Rev. Lett. 60 (1988) 1727–1730. [143] J. W. M. Frenken, B. J. Hinch, J. P. Toennies, C. Wöll, Anisotropic diffusion at a melting surface studied with He-atom scattering, Phys. Rev. B 41 (1990) 938–946. [144] M. Karimi, G. Vidali, I. Dalins, Energetics of the formation and migration of defects in Pb(110), Phys. Rev. B 48 (1993) 8986–8992. [145] L. D. Roelofs, E. I. Martir, in: The Structure of Surfaces III, S. Y. Tong, K. Takayanagi, M. A. Van Hove, X. D. Xie (eds.), Microscopic kinetics of the (1 × 2) missing row reconstruction of the Au(110) surface, (Milwaukee, Wisconsin, 1991), 248–252.

5

Diffusion on two-dimensional surfaces

In this chapter, we continue the task started in the last one: we will list diffusion characteristics determined on a variety of two-dimensional surfaces. For better orientation, ball models of fcc(111) and (100) planes have already been shown in Fig. 3.3, together with the (110), (100), and (111) planes of the bcc lattice in Fig. 3.4. In experiments it has been observed that on fcc(111) planes there are two types of adsorption sites: fcc (sometimes called bulk or stacking) and hcp (also referred to as surface or fault sites); these sites, indicated in Fig. 3.3b, can have quite different energetic properties. Two types of adsorption sites also exist on bcc(111) and hcp(0001) structures. However, on bcc(110) as well as on bcc(100) and fcc(100) planes, only one type of adsorption site has so far been observed, which makes it easier to follow adsorption on these surfaces.

5.1

Aluminum: Al(100) Experimental work on two-dimensional surfaces of aluminum was long in coming, and was preceded by considerable theoretical work, with which we therefore begin here. The first effort, by Feibelman [1] in 1987, was devoted to the Al(100) surface, and relied on local-density-functional theory (LDA–DFT). The primary aim of the work was to examine the binding energy of atom pairs, but he also estimated a barrier of 0.80 eV for the diffusive hopping of Al adatoms. Two years later, Feibelman [2] investigated surface diffusion which takes place on Al(100) by exchange of an adatom with one from the substrate. His findings are often cited as the first evidence for the occurrence of the exchange mechanism on a two-dimensional surface, but this process had previously been shown to occur in simulations by DeLorenzi and Jacuzzi in 1985 [3] on a bcc(100) plane of sodium. Feibelman arrived at a barrier to atom jumps over the surface of 0.65 eV, compared with 0.20 eV for diffusion by exchange of an aluminum adatom with an atom from the substrate lattice. The value for hopping differed from his previous findings. Liu et al. [4] in 1991 resorted to the embedded atom method (EAM) in order to evaluate the self-diffusion characteristics. They used two parametrizations: Adams, Foiles, and Wolfer [5] (AFW) and Voter–Chen [6]. The exchange mechanism on Al(100) was investigated but yielded contradictory results. With AFW parameters, ordinary hopping was more favorable, with a diffusion barrier of 0.4 eV and a prefactor Do = 2 × 10−3 cm2/sec, compared with a barrier of 0.69 eV for exchange. Using VC potentials the

262

Diffusion on two-dimensional surfaces

(b) 0.7 Activation Barrier ED (eV)

(a)

0.6 0.5 0.4

New

0.3 0.2

Exchange

0.1 0.0 4.05

Fig. 5.1

Hopping

4.10 4.15 4.20 Lattice Constant (Ang.)

(a) Model of adatom diffusion on fcc(100) plane, in which adatom enters the substrate and eventually displaces a lattice atom some distance away from the point of entry. (b) Activation energy for Al by three different diffusion mechanisms on Al(100) plane, as a function of the lattice constant. Dependence of hopping process is the opposite of the other two (after Cohen [7]).

exchange process was favorable over hopping, with an activation energy of 0.25 eVand a prefactor of Do = 4 × 10−2 cm2/sec compared with 0.46 eV for hopping. Calculations were also made with Lennard-Jones interactions and yielded a barrier of 0.60 and a prefactor of 1.0 × 10−2 cm2/sec for hopping, while for exchange the barrier was much higher, 2.28 eV and a prefactor of 9 × 10−2 cm2/sec. Three years later the same system was examined by Cohen [7] using EAM potentials. The activation energy for hopping was higher than for exchange in this system, 0.45– 0.58 eV for hopping and 0.1–0.24 eV for exchange, depending on the lattice constant, as in Fig 5.1b. She made interesting observations about the mechanism for diffusion, other than hopping or atom exchange, which also seem to have been seen earlier by Black and Tian [8]. In molecular dynamics simulations at 500 K, an adatom embedded in the lattice, and moved down the binding sites in the surface in an fcc row and over one, as shown in Fig. 5.1. The barrier was much higher (around 0.39–0.55 eV) than for standard atom exchange so the new mechanism of movement starts to be active at temperatures around half the melting point of the substrate. The same year the self-diffusion mechanism of aluminum adatoms was probed in molecular dynamics simulations by Gravil and Holloway [9] using effective medium potentials. They found that atom exchange with a barrier of only 0.271 eV predominated over simple atom hopping over a barrier of 0.692 eV because covalent bonding stabilized the intermediate state. The same year Stumpf and Scheffler [10] used density-functional theory to look at self-diffusion on aluminum surfaces and on the (100) plane they found an activation energy of 0.35 eV for exchange. Calculations with the discrete variational method in the local-density-functional approximation were made in 1997 by Li et al. [11]. For diffusion by atom exchange along they calculated a barrier of 0.27 eV, while atom hopping in the same direction required much more energy, 0.81 eV. The barrier to bridge hopping along

5.1 Aluminum: Al(100)

Fig. 5.2

263

Plot of self-diffusion barriers by atom hopping versus bond energy in bulk (equal to one sixth the cohesive energy) (after Feibelman [13]).

occurred over a barrier of 0.74 eV, making the exchange process the lowest energy event. A year later, Valkealahti and Manninen [12] performed calculations with EMT potentials for aluminum atoms migrating on polyhedral clusters using molecular dynamics. On a (100) surface, the activation energy for hops amounted to 0.30 eV. Exchange was observed only above 500 K, but it must be emphasized that these are values on clusters. Feibelman [13], in 1999, calculated the barrier to hopping of aluminum atoms, this time with the generalized gradient approximation, and found a value of 0.51 eV. He proposed that the barrier for hopping over the surface should be roughly one sixth the cohesive energy of the bulk and, as shown in Fig. 5.2, demonstrated that this worked quite well for self-diffusion barriers estimated in the generalized gradient approximation. A similar approach had previously been used by Wang [14] in 1989, who compared the measured activation energies for diffusion on W(211) to sublimation energies in an attempt to achieve insight into the diffusion process, and also by Kief and Egelhoff [15]. However this rule seems not to work for self-diffusion on Re(0001) compared to self-diffusion on W(110) [16] nor for diffusion by the exchange mechanism. The Ercolessi–Adams glue (EAG) potential [17] was used by Trushin et al. [18] in calculations of aluminum atom diffusion to obtain an energy barrier of 0.53 eV for atom hopping over the surface and 0.56 eV for atom exchange. The difference in energy was so small that they could not predict which mechanism was the favorable one. They also observed more complicated mechanisms, like three-atom exchange, with an activation energy of 0.85 eV and multi-particle exchange processes with barriers higher than 1 eV. Also in 2001, Ovesson et al. [19] carried out density-functional calculations for selfdiffusion by hopping and using transition-state theory they found an energy barrier of 0.57 eV. Papanicolaou et al. [20] made estimates relying on the second-moment approximation to tight-binding theory, that is RGL. The barrier for hopping was very low, 0.20 eV, and the prefactor amounted to 2.4 × 10−3 cm2/sec. They claimed that, as

264

Diffusion on two-dimensional surfaces

Al/Al(111)

Diffusivity D ( × 10–4 cm2/sec)

Diffusivity D ( × 10–4 cm2/sec)

10.0

1.0 ED = 0.067 eV

Fig. 5.3

1.00 hopping ED = 0.02 eV 0.10 exchange ED = 1.11 eV

0.01

(a) 0.1 10

Al/Al(100)

20

30 40 50 1/kT (1/eV)

60

(b) 15

20 25 1/kT (1/eV)

30

Arrhenius plots for self-diffusion on aluminum surfaces, examined by molecular dynamics simulations with RGL potentials. (a) Diffusion on Al(111) surface. (b) Diffusion both by hopping and atom exchange on Al(100) plane (after Papanicolaou et al. [20]).

illustrated in Fig. 5.3b, hopping was dominant up to 800 K. The barrier to diffusion by atom exchange was by comparison huge, amounting to 1.11 eV with a huge prefactor of 9.4 × 102cm2/sec, different from previous studies where exchange was the leading mechanism on this plane. In 2002, Fordell et al. [21] resorted to ab initio calculations with the VASP code [22–24] using the nudged elastic band method implemented for self-diffusion. In the local-density approximation [25], jumping occurred over a barrier of 0.52 eV; with the generalized gradient method (GGA) it was 0.50 eV. The barrier to atom exchange proved to be considerably lower for both approximations: 0.23 eV for LDA and 0.14 for GGA. At roughly the same time, Agrawal et al. [26] carried out detailed calculations for diffusion, without any concern for many-atom effects. They worked with LennardJones potentials to do Monte Carlo simulations for several crystal faces of fcc metals. On the Al(100), the values were 0.60 eVand 6.1 × 10−3 cm2/sec. It must be noted here that exchange was not considered at all as a mechanism of diffusion. The value reported is out of line. A year later, density-functional theory estimates were carried out by Sun and Li [27] using the Vienna ab initio package with the generalized gradient approximation and a 4 × 4 surface unit cell for self-diffusion of aluminum. The diffusion occurred by exchange, with an activation energy of 0.17 eV; hopping required overcoming a much larger barrier of 0.46 eV. These are values comparable with previous findings of Fordell et al. obtained with the same method. Chang and Wei [28] in 2005 again used ab initio density-functional theory. For hopping they arrived at a barrier of 0.67 eV; for diffusion by atom exchange the barrier was much lower, 0.13 eV. They also explored the effect of the size of the unit cell used in the calculations on the activation energy, as previously done by Perkins and DePristo [29]

5.2 Aluminum: Al(111)

265

and found that for exchange the unit cell size can have a huge effect on the magnitude of the energy barrier. For exchange, a (5 × 5) cell was desirable to ensure relaxation and to get values comparable with experiment. Diffusion on aluminum surfaces has received considerable theoretical attention, but not much experimental work has been reported examining atom movement. Only theoretical results are available for self-diffusion on Al(100) – there have been no experiments done – and these results range quite widely from 0.2 eV to 0.8 eV for hopping and from 0.14 eV to 0.69 eV for exchange. There is no unanimity about the mechanism of diffusion, whether it occurs by hops or in an exchange process, but exchange over a barrier around 0.25 eV seems to be a likely event. Additionally, there is a possibility that long exchange events will occur on this surface at higher temperatures. No investigations of hetero-diffusion on aluminum, either experimental or theoretical, have been done so far.

5.2

Aluminum: Al(111) Compared to the Al(100) surface, the Al(111) is much smoother and has two kinds of adsorption sites, fcc and hcp, shown in Fig. 3.3b. These may have different energetic characteristics. Investigations on this plane also started with theoretical studies, and experimental work caught up a bit later. Theoretical studies of self-diffusion began with the embedded atom method (EAM) work of Liu et al. [4] in 1991. Using the parametrization of Adams, Foiles, and Wolfer [5] (AFW) they obtained a barrier of 0.074 eV and Do = 9 × 10−4 cm2/sec; using the Voter– Chen [6] approximation (VC), the barrier was lower, 0.054 eV and Do = 1.6 × 10−3 cm2/ sec. An estimate relying on the Lennard-Jones potentials came out with a diffusion barrier of 0.11 eV. Liu et al.’s efforts were followed by Stumpf and Scheffler [10,30] in 1994 with density-functional theory calculations. For normal hopping they reported an activation energy of 0.04 eV for jumps between hcp sites on the surface. They claimed that for Al the hcp site is a stable binding site, with a difference in energy between hcp and fcc site of 0.04 eV. DFT was also used in 1998 by Bogicevic et al. [31] to determine the same diffusion barrier of 0.04 eV. The same year, Valkealahti and Manninen [12] did calculations with EMT potentials for aluminum atoms migrating on polyhedral clusters using molecular dynamics. On an Al(111) surface of a polyhedral, the activation energy for hops proved to be 0.02 eV. So far only Liu et al. [4] bothered to determine prefactors for the diffusivity; later studies gave only activation energies. The determinations from DFT are exactly the same; EAM seems to give a slightly higher value, but also not far away. At this stage there began the first efforts at experimental determinations for diffusion on Al(111) by Barth et al. [32], using the STM to determine the saturation density of islands. From measurements at eight temperatures, from 60 to 180 K, shown in Fig. 5.4, they derived an activation energy for aluminum self-diffusion of 0.042 ± 0.004 eV, in reasonable agreement with previous theoretical work. Their prefactor, however, had the unusually small value of Do = 2 × 10−9 ± 0.25 cm2/sec. The authors attributed this to a general effect, which lowers the prefactor when diffusion takes place in “the weak

266

Diffusion on two-dimensional surfaces

Temperature T(K) 125 83

250

62.2

nx (islands/site)

(a) 10–3

(b)

10–4

10–5

4

8

12

16

1000/T (K–1)

Fig. 5.4

Aluminum islands on Al(111). (a) Plot of aluminum island density on Al(111) against reciprocal temperature. Solid circles: Al evaporated from BN tube, 0.11 ML coverage, deposition rate 7.2 ± 0.7 × 10−3 ML/sec. Open circles: Evaporation from Al2O3 tube, 0.10 ML, 1.05 ± 0.06 × 10−2 ML/sec. Full squares: results of Barth et al. [32]. (b) STM image after deposition at 183 K (after Michely et al. [33]).

corrugation regime”, over a barrier of less than 0.1 eV. The low value of the prefactor is quite suspicious, however, and leads to concerns about the experiments and about the effect of atomic interactions on the underlying nucleation theory. The findings of Barth et al. [32] were soon challenged by Michely et al. [33], who repeated island density measurements on Al(111), with the aluminum evaporated both from an Al2O3 tube as well as a tube of boron nitride; comparison with results as well as data from Barth et al. [32] is shown in Fig. 5.4. The second arrangement gave a much cleaner deposit, but yielded results quite different from what had been reported by Barth et al. [32]. Although they recognized their experiments still needed further work, Michely et al. reported an activation energy for diffusion on clean Al(111) of 0.06 eV, and a frequency prefactor of ν0 = 1 × 1011 sec−1. They suspected that the low value of the prefactor obtained by Barth et al. was connected with coadsorption of impurities rather than with the weak corrugation of the potential energy of the adatom. Later, in 2003, Busse et al. [34] made an estimate of 0.04 eV for the diffusion barrier, with a prefactor of 5 × 1011 sec−1 again based on measurements of the island density in Fig. 5.5, so that at least the activation energies proposed by the two groups are roughly the same. Chang et al. [35] used density-functional calculations for self-diffusion on Al(111) and found again a barrier of 0.04 eV; with embedded atom potentials the result was similar, 0.03 eV. Ovesson et al. [19] once more carried out density-functional calculations for adatom movement by hopping. Using transition-state theory, they found an energy barrier of 0.042 eVand a frequency prefactor of 4 × 1012 sec−1. They attributed the low prefactor obtained in the experiments by Barth et al. [32] to long-range interactions neglected in nucleation theory rather than to the adsorption of impurities claimed by Michely et al. [33]. A substantial number of diffusivities have been derived from mean-field nucleation theory by plotting the logarithm of the island density at saturation against the reciprocal

267

5.2 Aluminum: Al(111)

10–2

nx (islands/site)

I 10–3 II II 10–4

10–3

III 10–5

III

10–4 10–3

10–6

Fig. 5.5

100

200

300 400 Ecoh /kT

10–2 10–1 Ff (ML/sec)

500

600

Arrhenius plot for island density nx on Ir(111) (solid squares) and Al(111) (solid triangles). Inset gives island density versus rate of deposition on Ir(111) (after Busse et al. [34]).

–1

10log[n ] x

–2

–3

–4

KMC – with interaction NL–MFNT – with interaction KMC – no interaction NL–MFNT – nointeraction

–5 5

10

15

20

ED /kT Fig. 5.6

Saturated density of aluminum islands on Al(111) in its dependence upon the reciprocal temperature at 0.05 coverage and an impingement rate of 0.03 ML/sec. Data from simulations by kinetic Monte Carlo methods (after Ovesson [36]).

temperature. However, this approach has come under suspicion, as classical nucleation theory ignores interactions between metal atoms which are known to be strong. Ovesson [36] has therefore worked out a nucleation theory allowing for adatom–adatom interactions. A comparison of the island density plotted against the reciprocal temperature both with and without interactions is shown in Fig. 5.6, where the mean-field approximation is also tested against kinetic Monte Carlo simulations. Markedly larger densities are obtained with interactions (patterned on results for Al on Al(111)) than without. It will

268

Diffusion on two-dimensional surfaces

be interesting to see if this procedure can be tested for ordering diffusivities in interacting systems. A slightly different approximation was undertaken by Papanicolaou et al. [20], who made estimates of the diffusion of aluminum atoms on the Al(111) plane relying on the second-moment approximation to tight-binding theory (RGL). From the Arrhenius plot shown in Fig. 5.3a. they arrived at an activation energy of 0.067 eV with a prefactor of Do = 1.4 × 10−3 cm2/sec. This value is the result of averaging the two barriers corresponding to fcc and hcp positions. However, from a trajectory analysis, only single hops were detected. Agrawal et al. [26] worked with Lennard-Jones potentials to do Monte Carlo simulations for several crystal faces of fcc metals. For aluminum adatoms diffusing on the (111) plane they found an activation energy of 0.13 eV and a prefactor Do = 3.2 × 10−4 cm2/sec. This study shows that pair potentials are not suitable to describe movement on this plane. Density-functional theory estimates using the Vienna ab initio package have been carried out again for diffusion of aluminum on Al(111) planes by Sun and Li [27]. Aluminum adatoms jump across the bridge between two substrate atoms with an activation energy of only 0.04 eV. Exchange on this plane involved a large barrier of 0.63 eV, making it unlikely to happen. Polop et al. [37] as well as Busse et al. [34] reported further work on the saturation island density of Al on Al(111), but came to the conclusion that because of uncertainties in the diffusivity of aluminum dimers, the experiments did not allow an unequivocal decision about the diffusion characteristics of single aluminum atoms. They adopted the value of 0.04 eV for the diffusion barrier, obtained in density-functional calculations, with a frequency prefactor of 5 × 1011 ± 0.5 sec−1. The last work on Al(111) was reported by Polop et al. [38] in 2005. They looked at the island density under carefully controlled clean conditions, but also in the presence of contamination. Island nucleation on the clean surface as well as on a surface with oxygen clusters is compared in Fig. 5.7. For clean Al(111) the activation energy for self-diffusion was found as 0.07 ± 0.01 eV, with a Temperature T(K) 10–2

250

nx (islands/site)

1

167

125

100

83

71

10

12

14

2

10–3 10–4 10–5 10–6

i=1

i=2 4

6

8

1000/T (K–1) Fig. 5.7

Density of aluminum islands on Al(111) as a function of reciprocal temperature, grown on clean surface (solid squares) and on surface with 0.024 ± 0.002 oxygen clusters. On the clean surface the activation energy for diffusion was 0.07 ± 0.01 eV (after Polop et al. [38]).

5.4 Iron: Fe(100), (111), (110)

269

frequency prefactor ν0 = 5 × 1011 ± 0.5 sec−1. On the surface contaminated with oxygen the diffusion barrier decreased to 0.03 eV and the prefactor was 8 × 106 sec−1. Since in previous experiments by Barth et al. [32] the activation energy was 0.042 ± 0.004 eV, with a frequency prefactor of 8 × 106 ± 0.25 sec−1, Polop et al. took the low values reported for the diffusion barrier and the prefactor as a general indication of contamination. The activation energies calculated for self-diffusion on Al(111) scatter between 0.02 eV and 0.07 eV, excluding the results of Agrawal et al. [26] which are completely out of range. Experimental studies of the diffusion barrier are in the same range, from 0.042 eV to 0.070 eV, with the latter experimentally preferred. Direct measurement of adatom displacements would also be very desirable. The experimental findings are in good agreement with most of the calculational results. There is no information about hetero-diffusion on this plane.

5.3

Potassium In 1997 Fuhrmann and Hulpke [39] did helium scattering experiments on a Ni(100) surface covered with 4 ML of potassium. The activation energy for diffusion of single potassium atoms on such a layer was derived as 0.063 ± 0.015 eV, with a prefactor of Do = (9.5 ± 0.9) ×10−4 cm2/sec. At 300 K, desorption contributed to the measurements but single jumps were favored in diffusion on the surface. On lowering the temperature to 285 K, the percentage of long jumps increased; jumps covering two or three spacings were observed, quite a surprising result since the presence of long jumps is usually associated with increased temperatures. Fuhrmann and Hulpke also claimed that their diffusion barrier was an average value from a number of different crystallographic directions, since they detected a rich distribution of domains in the system. It is difficult to judge how their results correspond to single atom movement, as well as the possible influence of strain and defects on movement.

5.4

Iron: Fe(100), (111), (110) Measurements of the self-diffusion of iron on Fe(100) were made by Stroscio et al. [40,41], using the scanning tunneling microscope and reflection high energy electron diffraction. They measured the temperature dependence of the density of islands formed in the temperature range 20–250 °C. The data were taken at a starting coverage of 0.07 ± 0.016 monolayers and interpreted according to classical nucleation theory; it is shown in Fig. 5.8. It was assumed that only individual adatoms can diffuse in the investigated temperature range and that the capture number for islands was unity; this assumption could affect the diffusivity by a factor less than ten. They arrived at an activation energy for diffusion of 0.45 ± 0.08 eV, with a prefactor of Do = 7.2 × 10−4 cm2/sec. Pfandzelter et al. [42] later studied the submonolayer growth of iron on Fe(100) by examining the scattering of a grazing angle beam of 25 keV He+ ions. From the temperature dependence of the saturation density of islands, shown in Fig. 5.9, they

270

Diffusion on two-dimensional surfaces

(a)

(b)

Temperature (°C) 227 127 60

13

10–6 10–7

1013 10–8 10–9 1012 10–10 10–11 11

10

1.5

2.0

2.5

Diffusion Constant (cm2/sec)

Island Density (cm–2)

394

(c)

10–12 3.5

3.0

1000/T (K–1) Fig. 5.8

Growth of iron on Fe(100) surface. STM images of single layer islands grown at (a) 20 °C, (b) 108 °C, and (c) 163 °C. (d) Arrhenius plot of the island density and the diffusivity for Fe on Fe(100). Coverage 0.07 ± 0.016 ML, at a flux of 1.4 ± 0.3 × 1013 atoms/cm2 sec (after Stroscio et al. [40]).

Temperature T(K) 667

Island Density (cm–2)

1013

500

400

333

286

Ion scattering STM

1012

1011 ED = 0.495±0.50 eV ν0 = 9 × 1011±0.7 sec–1

1010 1.5

2.0

2.5

3.0

3.5

1000/T (K–1) Fig. 5.9

Saturation density of Fe islands on Fe(100) in its dependence upon reciprocal temperature. Open squares: STM results from Stroscio. Solid circles: from ion scattering (after Pfandzelter et al. [42]).

271

5.4 Iron: Fe(100), (111), (110)

Table 5.1 Temperature variation of the island--island separation Li determined by RHEED and STM [43]. T (K)

Li(RHEED) (Å)

Li(STM) (Å)

340 375 430 490 600

40 48 68 162 292

38 47 66 162 291

667

Temperature T (K) 500 400 333

286

5.5

log

5.0 4.5 4.0 3.5 3.0 1.5

Fig. 5.10

ED = 0.37 eV

2.0

2.5 1000/T (K–1)

3.0

3.5

Dependence of the separation of iron islands on Fe(100) upon the reciprocal temperature, yielding an activation energy of 0.37 eV at low temperatures (after Dulot et al. [43]).

derived an activation energy of 0.495 ± 0.050 eV and a frequency prefactor of 9 × 1011 ± 0.7 sec−1 for diffusion of iron atoms, in good agreement with the prior work of Stroscio et al. [40]. In 2003, Dulot et al. [43] examined the deposition of iron on Fe(100) using RHEED as well as STM, and were able to determine the island density achieved in growth at substrate temperatures ranging from 340 K to 600 K. They deduced the island–island distance Li by analyzing the shape of diffraction peaks in RHEED, as well as measuring the distance with scanning tunneling microscopy STM. The data are presented in Table 5.1, and the agreement is excellent. From the temperature dependence of the island separation shown in Fig. 5.10, they arrived at an activation energy for surface diffusion of 0.37 eVat temperatures below 450 K. The agreement with the work of Stroscio et al. [40] is not too bad. In the year 2000, Köhler et al. [44] performed STM measurements as well as Monte Carlo simulations with Finnis–Sinclair [45] potentials to better understand homoepitaxy on the bcc Fe(110) plane. As part of this effort they also calculated the energy barriers to atom diffusion. For motion by jumps on the rigid surface the barrier in the

272

Diffusion on two-dimensional surfaces

direction was 0.38 eV, and along , over a surface atom, it proved to be much higher, 0.88 eV. On fully relaxing the iron substrate, the barrier to diffusion along became much smaller, amounting to only 0.27 eV. Seven years later, Chan et al. [46] looked at the movement of Fe with EAM potentials and MD simulations. Single adatom movement proceeded along directions with an energy of 0.4 eV and a prefactor 1.27 × 10−2 cm2/sec. The potential barrier is symmetric, with a saddle point at 0.3 eV, and has a local minimum 0.02 eV deep in the middle between two stable adsorption places. Note the disagreement between the energy given for the movement of monomer and the height of the barrier, about which the authors did not comment. The data turn out to be comparable with the findings of Köhler et al. [44]. Unfortunately, no experimental estimates of the barrier on Fe(110) appear to be available. No good theoretical estimates for self-diffusion of iron have been made, but Flahive and Graham [47], using Morse potentials, calculated a barrier of 1.4 eV for self-diffusion on Fe(111), 0.6 eV on Fe(100), and ~0.25 eV for Fe(110), as shown in Fig. 4.9. Kief and Egelhoff [15] much later made approximate determinations by scaling the barrier for other systems with the cohesive energy. For self-diffusion on Fe(111) they obtained a barrier of 0.18 eV by comparing the iron surface with Ir(111) [48], for which the diffusion energy was known. Since the structure of the Ir(111) plane (a very smooth surface) is quite different from that of Fe(111) (a bcc metal and quite rough), such estimates can be misleading. To obtain the activation energy for iron diffusion on Fe(100) as 0.46 eV, scaling from copper self-diffusion on Cu(100) was used. In 1999, Davydov [49] estimated the activation energy for self-diffusion on the Fe(110) surface as 0.65 eV, based on his cohesion approach. More reliable estimates of diffusion characteristics were done by Spisa´k and Hafner [50,51] in 2004. They looked at the movement of Fe atoms on 1 ML of Fe covering a W(100) surface with first-principles VASP calculations. The shortest bridge hopping along cost 0.4 eV. For jumps to next-nearest-neighbor sites along the direction the barrier was only 0.5 eV. Exchange was a bit more expensive, requiring 0.7 eV. It should, however, be mentioned that 1ML of iron on a W(100) surface is very likely influenced by strain, which will change the diffusion. The authors did not discuss this factor. Self-diffusion on the Fe(100) plane has been investigated using molecular dynamics simulations by Chamati et al. [52]. To perform this study, the authors first constructed a new EAM potential, which was fitted to measured quantities as well as first-principles calculations both for bcc α-iron and fcc γ-iron. From analyses of atom trajectories in the temperature range 600–950 K they found that atom movement took place by three mechanisms: diffusion by diagonal exchange, as in the first row of Fig. 5.11, by atoms hopping as in the second row, and finally non-diagonal exchange, shown in the third row of Fig. 5.11. The temperature dependence of all three processes is shown in Fig. 5.12. For diagonal exchange the barrier was 0.66 eV, with a prefactor of 113 × 10−3 cm2/sec; for hopping a barrier of 0.92 eV was found with a prefactor of 206 × 10−3 cm2/sec, and finally for non-diagonal exchange an activation energy of 0.99 eV with a prefactor of 306 × 10−3 cm2/sec was calculated. Using the nudged elastic band (NEB) method, Chamati et al. also calculated static barriers of 0.60 eV, 0.84 eV, and 0.97 eV for diagonal exchange, atom hopping, and non-diagonal exchange. They also observed long and correlated jumps as well as diffusion by concerted exchange, but the frequency was

273

5.4 Iron: Fe(100), (111), (110)

Fig. 5.11

Atom trajectories of Fe atoms during diffusion on Fe(100) plane deduced in EAM simulations. Top row: diagonal atom exchange. Middle row: diffusion by atom hopping. Bottom row: non-diagonal exchange mechanism (after Chamati et al. [52]). 103

Diffusivity D(10–7cm2/sec)

Fe/Fe(100) 102

Diagonal exchange 101

Jump

Non-diagonal exchange 100

10–1 10

12

14

16

18

20

1/kT (eV–1) Fig. 5.12

Dependence of Fe atom diffusion upon reciprocal temperature for three mechanisms of diffusion. Diagonal atom exchange has the lowest activation energy of 0.66 eV (after Chamati et al. [52]).

too low to arrive at the characteristics of these processes. It must be noted, however, that all the results are too high compared with what has been obtained in experiments. Recently, Shvets et al. [53] looked at nanowedge formation of iron on Mo(110). As shown in Fig. 5.13, they calculated a hopping barrier for Fe on Fe(110) at different strains

274

Diffusion on two-dimensional surfaces

Diffusion Barrier ED (eV)

0.8

0.6

0.4

0.2

0.85

Fig. 5.13

0.90

0.95 1.00 Lattice constant

1.05

1.10

Effect of strain on barrier to diffusion of Fe on Fe(110) (after Shvets et al. [53]).

using density-functional theory in the GGA approximation, relying on the CASTEP numerical code to obtain diffusion barriers. They worked with a four layer thick slab and a 2 × 2 cell – two layers were fixed and two relaxed. The barrier was around 0.65 eV on the surface without strain. Their arrangement did not take into account the presence of the Mo substrate underneath the Fe layer, and the simulation settings were very limited. It is unlikely they obtained reliable values for this system. Iron (111) and (110) definitely need more work, both experimental and theoretical. For iron (100), experiments seem to be in agreement, but theory still needs effort. There is no data on hetero-diffusion for these substrates.

5.5

Nickel: Ni(111)

5.5.1

Self-diffusion Tung and Graham [54] were the first to undertake a technically quite difficult project – examination of self-diffusion on nickel surfaces using the field ion microscope, with the surface cooled to ~20 K. On the (111) planes, schematics of which were shown in Fig. 3.3, activation energies for diffusion were derived from the onset temperature, assuming the prefactor to be (kT/2h)‘2 ; here k stands for Boltzmann’s constant, h for Planck’s constant, and ‘ for the jump length. This yielded a somewhat uncertain value of 0.33 eV for the barrier on the (111) surface. They also looked at adsorption places on the Ni(111) surface and definitely recognized two types, with slightly different binding energies; both sites were occupied after thermal treatment. Twenty years later, Fu and Tsong [55] again looked at Ni(111), also with FIM but with a sample pre-annealed before tip formation. With the sample cooled to 30 K during measurements they derived an activation energy of 0.22 ± 0.02 eV and a prefactor Do = 1 × 10−5.2 ± 0.7 cm2/sec, based on measurements at four temperatures shown in Fig. 5.14. The agreement between the two studies is acceptable, taking into account

275

5.5 Nickel: Ni(111)

–37 ED = 0.22±0.02 eV Do = 1 × 10–5.2±0.7cm2/sec

ED = 0.59±0.03eV Do = 1 × 10–5.2±0.6cm2/sec

–38 In[/4t]

In[/4t]

–38

–40

–39

–40 (a)

–42

–41 10

11 1000/T (K–1)

Fig. 5.14

12

(b) 3.8

4.0

4.2

1000/T (K–1)

Arrhenius plots for self-diffusion of Ni adatom on nickel, as measured in the field ion microscope (FIM). (a) On Ni(111) plane. (b) On Ni(100) surface. Insets give indication of diffusion mechanism (after Fu et al. [55]).

the preliminary character of the first studies. It should be noted that no attempt to determine the binding sites on Ni(111) was made. Although there have been only two experimental studies of the diffusion characteristics of nickel atoms on nickel (111), more than ten efforts have been made to calculate these diffusivities. Flahive and Graham [47], using Morse potentials, calculated a barrier for selfdiffusion on Ni(111) of ~ 0.01 eV, as in Fig. 4.9, relatively far away from experimental determinations. The first effort with many-atom interactions was done by Liu et al. [4] in 1991, and involved embedded atom calculations for atom hopping, using both AFW [5] and VC [6] parametrizations. They obtained a barrier of only 0.056 eV and a prefactor of 5 × 10−4 cm2/sec with AFW, and 0.063 eV and 6.2 × 10−4 cm2/sec with VC parameters. These values are again considerably smaller than what has been found in the experiments. Liu et al. also used a Lennard-Jones potential for estimates. On the (111) plane they found a barrier of 0.15 eV with a prefactor of 9.0 × 10−4 cm2/sec. Comparisons of the diffusion barriers on nickel surfaces were obtained by Rice et al. [56] with different EAM potentials. Rates at different temperatures were calculated using transition-state theory with the FBD [57] as well as the VC approximations, but also with the potentials of Oh and Johnson [58] as well as of Ackland et al. [59]. Static estimates were made, giving values of 0.06 eV with FBD parameters, 0.07 eV with VC, 0.05 eV for OJ, and 0.05 eV for ATVF. All these values are much too low compared with experiments. Shiang [60] performed molecular dynamics estimates using the Sutton–Chen (SC) and also the Morse potential to estimate the self-diffusion characteristics of Ni atoms. Using the SC potential he arrived at an activation energy of 0.086 eV and a prefactor Do = 8.74 × 10−4 cm2/sec from an Arrhenius plot of the rate of diffusion at different temperatures, shown in Fig 5.15. From calculations with Morse potentials, which are also shown in Fig. 5.15, the values for the activation energies were 0.016 eV and for the prefactor 5.04 × 10−4cm2/sec. It is clear that many-body interactions play an important role, the activation energies being much higher with Sutton–Chen potentials.

276

(a)

Diffusion on two-dimensional surfaces

–9

(b) –7.6 Ni on Ni(100)

Ni on Ni(100)

–10

–7.8

–11

–8 In D

In D

–12 –13

–8.4

–14

Morse potential SC potential

–15 –16 150 200

Fig. 5.15

–8.2

250

300 350 1/T

Morse potential SC potential

–8.6 400

450

500

–8.8 150

200

250

300 350 1/T

400

450

500

Arrhenius plots for Ni diffusion on nickel surfaces. (a) On Ni(100): ED = 0.530 eV, Do = 2.87 × 10−3 cm2/sec from many-body potential, and 0.298 eV from Morse potental. (b) On Ni(111): ED = 0.086 eV, Do = 8.47 × 10−4 cm2/sec from many-body potential, and 0.016 eV from Morse potential (after Shiang et al. [60]).

By resorting to effective medium theory, Stoltze [61] arrived at an activation energy of 0.068 eV for self-diffusion. The value is again on the low side compared with experiment. In 1996, Li and DePristo [62] used corrected effective medium theory and molecular dynamics to calculate an activation energy for diffusion of 0.036 eV. Densityfunctional estimates were done by Mortensen et al. [63] using both the local-density and generalized gradient approximation; in the former they found a barrier of 0.16 eV, in the latter 0.11 eV, both on the static lattice. The uncertainty of the energy prediction was estimated as 0.02 eV. They also looked at the differences between fcc and hcp sites; the former seemed to be 0.05 eV stronger than hcp. Chang et al. [35] relied on embedded atom potentials to arrive at an activation energy of 0.06 eV for diffusion of nickel atoms on Ni(111). In 2000, Haug and Jenkins [64] examined the effect of hydrogen on epitaxial growth. They also looked at self-diffusion using EAM potentials [65,66] and found again a barrier of 0.05 eV for hopping between fcc sites of the plane. The presence of hydrogen lowered the barrier to 0.02 eV. The exchange mechanism required much higher energy of 2.15 eV and hydrogen reduced this barrier to 1.55 eV. It was still too high for this mechanism to participate during growth. Using the embedded atom approximation for potentials, Kürpick [67] has done transition-state calculations for self-diffusion on nickel and copper surfaces. For nickel she found a diffusion barrier of 0.063 eV, and a prefactor of (1.8 ± 0.1)×10−4 cm2/sec. Agrawal et al. [26] depended on Lennard-Jones potentials in their estimates for the hopping barrier on Ni(111) and obtained a value of 0.14 eV and a prefactor of 1.7 × 10−4cm2/sec, in very good agreement with previous findings with the same potential. More recently, Bulou and Massobrio [68] utilized molecular dynamics simulations in the second-moment approximation to tight-binding. For self-diffusion of Ni they arrived at a barrier of 0.049 eV for hopping, and a much higher value, in excess of 2 eV, for atom exchange. The barrier for hopping is again much smaller than the experimental value. Kong and Lewis [69] calculated the prefactor to self-diffusion on Ni(111) at 300 K relying on EAM potentials by summing the vibrational contributions over all atoms,

5.6 Nickel: Ni(100)

277

and found a value of 2.99 × 10−4 cm2/sec starting from an fcc site and 2.89 × 10−4 cm2/sec starting from an hcp site. These prefactors essentially did not change over the next 300 K. Kim et al. [70], using action-derived molecular dynamics [71] with RGL potentials, have recently estimated a self-diffusion barrier of 0.061 eV for nearest-neighbor jumps, and 1.633 eV for exchange, the former in agreement with previous calculations. Estimates for self-diffusion on Ni(111) are quite consistent. Exchange has been found to involve overcoming a very high barrier, and is therefore unlikely to occur. The activation energies calculated at different laboratories are usually reasonably close together. Different adsorption sites available on Ni(111) have been recognized. The most significant concern, however, is the disagreement between calculational efforts and experiments, which have yielded a much higher activation energy than the calculations. Additional experimental effort might be in order.

5.5.2

Hetero-diffusion Only one experimental investigation was done for hetero-diffusion on the Ni(111) plane. Diffusion of platinum atoms was examined on Ni(111) by Kellogg [72] using the field ion microscope. Platinum atoms were mobile on Ni(111) while imaging at 77 K, so little data was obtained, but there were no indications of motion by atom exchange. Kellogg suggested a diffusion barrier of 0.2–0.3 eV. Kim et al. [70] also made estimates with RGL potentials for hopping of a number of hetero-atom diffusion events on Ni(111). For Cu, the barrier was 0.050 eV, for Pd 0.045 eV, for Ag 0.038 eV, for Pt 0.018 eV, and for Au 0.038 eV. It must be noted that hetero-diffusion has not been carefully examined in theoretical calculations. Only for Pt on a Ni(111) surface are data available for comparison of experimental and theoretical barriers, and these are an order of magnitude apart: ~0.2 eV from experiment versus 0.018 eV from calculations. However, the value from FIM observed by Kellogg [72] is only an estimate, and might be too high. More thorough studies would certainly be advantageous for this system.

5.6

Nickel: Ni(100)

5.6.1

Self-diffusion Tung and Graham [54] examined a nickel surface using the field ion microscope, with the surface cooled to ~ 20 K. Imaging of the (100) planes, schematics of which were shown in Fig. 3.3, was quite challenging. Nevertheless, an activation energy for diffusion of 0.63 eV was derived from the onset temperature, assuming the prefactor to be (kT/2h)‘2 . Two decades later, Fu and Tsong [55], also used the FIM but pre-annealed their wire before electro-etching the tip for longer life and for removing impurities. With the sample cooled to 30 K they derived an activation energy of 0.59 ± 0.03 eV and a prefactor equal to 1 × 10−5.2 ± 0.6 cm2/sec, shown in Fig. 5.14b. The agreement between the two studies for the (100) plane is surprisingly good. Fu and Tsong also showed that diffusion

278

Diffusion on two-dimensional surfaces

[010]

[110]

[100] 250 K, 29 heating periods

250 K, 32 heating periods

Fig. 5.16

260 K, 30 heating periods

Maps of sites occupied by Ni atoms diffusing on Ni(100) plane, indicating diffusion by atom exchange (after Fu and Tsong [55]).

occurred by exchange between the adatom and a lattice atom; this was deduced from the map of sites visited by adatoms, displayed in Fig. 5.16. Work function changes on a Ni(100) surface were measured at different temperatures by Schrammen and Hölzl [73] after deposition of nickel atoms to reasonably low coverages (1/80 and 1/40 of a monolayer). Monte Carlo simulations were carried out to fit the measured work function increase with temperature and they found an activation energy for atom jumps between nearest-neighbor sites of 0.60 ± 0.02 eV, assuming a frequency prefactor of 1012 sec−1. This result is in excellent agreement with the field ion microscopic observations of diffusion on this plane. In 1995 Bartelt et al. [74] analyzed island densities on Ni(100), on which half a monolayer of nickel had been deposited at 300 K by Kopatzki et al. [75]; they found an activation energy for diffusion of more than  0.43 eV and a frequency prefactor of  1012 sec−1. There has been quite a lot of theoretical activity for predicting self-diffusion characteristic on Ni(100), starting with Flahive and Graham [47]. They used Morse potentials and calculated a barrier for self-diffusion on Ni(100) of ~ 0.3 eV, shown in Fig. 4.9, quite far from the experimental determinations. Many-atom interactions were taken into account by Liu et al. [4] in 1991, with embedded atom calculations using both AFW [5] and VC [6] parametrizations. They found for hopping an activation energy of 0.63 eV and a prefactor Do = 1.6 × 10−3 cm2/sec with AFW parameters; with VC parameters these values turned out higher, 0.68 eV for the barrier and 5.4 × 10−3 cm2/sec for the prefactor. The barrier to atom exchange came out to be 0.93 eV using AFW potentials, and 1.15 eV with a prefactor of 4 × 10−2 cm2/sec for VC interactions, much higher than the activation energy for hopping. These estimates do not reflect on the experimental findings about the mechanism. Liu et al. also used a Lennard-Jones potential for estimates and obtained a barrier of 0.80 eV, with a prefactor of 6.9 × 10−3 cm2/sec. Transition-state estimates using the corrected effective medium theory were done for hopping of nickel atoms by Sanders and DePristo [76], who found a barrier of 0.61 eV

5.6 Nickel: Ni(100)

279

with a prefactor of 3.2 × 10−3 cm2/sec. From molecular dynamics they arrived at a diffusion energy of 0.63 eV; exchange was not considered. Comparisons of the diffusion barriers on Ni(100) obtained with different EAM potentials were done by Rice et al. [56]. They used transition-state theory with the FBD and VC approximations, the Oh and Johnson potential (OJ) [58] as well as the potential of Ackland et al. (ATVF) [59]. The results were as follows: activation energy 0.64 eV, Do 3.5 × 10−3 cm2/sec; 0.70 eV, 3.7 × 10−3 cm2/sec; 0.45 eV, 3.5 × 10−3 cm2/sec; 0.93 eV, 6.4 × 10−3 cm2/sec, respectively. Values comparable with previous findings were obtained with the first two potentials. A comparison is presented in Fig. 4.10. What emerged from this study is that not all potentials describe nature correctly, and the scatter in the data obtained from different approaches can be quite big. Shiang [60] utilized molecular dynamics with the Sutton–Chen potential and estimated a barrier of 0.530 eV, with the prefactor equal to 2.87 × 10−3 cm2/sec. He compared this finding with data from Morse potentials, plotted in Fig. 5.15; the value for the activation energy was 0.298 eV with a prefactor of 1.06 × 10−3cm2/sec. As already shown for Ni(111), many-body interactions play an important role for Ni(100) as well, the activation energies being much higher with Sutton–Chen potentials. Shiang did not consider exchange as a possible mechanism of movement. One year later Perkins and DePristo [77] also made estimates with the effective medium theory to describe inter-atomic energies. They found an activation barrier of 0.68 eV for hopping compared with 0.65 eV for atom exchange, but in estimates using MD/MC-CEM reached a lower value of 0.62 eV for hopping and 1.4 eV for exchange. They reanalyzed this data a year later [29], allowing the number of movable atoms in the lattice to vary. The hopping barrier from effective medium theory remained essentially constant, varying from 0.68 eV to 0.67 eV as the number of movable atoms rose from 13 to 226. For exchange, however, the activation energy changed dramatically from 0.65 eV to 0.47 eV. Similar effects were found in calculations with MD/MC-CEM: the hopping barrier changed from 0.62 eV to 0.60 eV, and for exchange from 1.4 eV to 1.0 V. From this investigation it is not clear which mechanism is favorable, jumping or exchange, since two methods contradict each other. In molecular dynamics simulations with embedded atom potentials, Cohen [7] found long atom exchange events in self-diffusion on a number of metals, among them Ni(100). The barrier to these long exchanges was 1.42 eV; for ordinary hopping the activation energy was much lower, 0.74 eV, and for the usual atom exchange 0.85 eV. Exchange could occur at elevated temperatures such as 1000 K, as the barriers for hopping and exchange are reasonably close together, suggesting the coexistence of the mechanisms. The same year, Stoltze [61] utilized the effective medium theory to arrive at an activation energy of 0.558 eV for atom hopping; he did not consider exchange on this plane. Boisvert et al. [78] did molecular dynamics simulations based on embedded atom potentials for nickel atoms jumping and undergoing exchange; calculations were done on a layer of 72 atoms. For atom hopping they arrived at a barrier of 0.67 eV, with a frequency prefactor of 3.71 × 1012 sec−1. For atom exchange the barrier was much higher, 1.29 eV, with a frequency prefactor of 7.28 × 1012 sec−1. The latter barrier may have been affected by the small size of the substrate surface, but the activation energies are so

280

Diffusion on two-dimensional surfaces

different that jumping should still be the favored mechanism. Shi et al. [79] also relied on embedded atom potentials to evaluate the activation energies for self-diffusion on three fcc(100) planes, with 12 layer slabs consisting of 128 atoms in each layer and three bottom ones fixed. For single atom movement on Cu(100), they found a barrier of 0.503 eV, on Ag(100) 0.478 eV, and on Ni(100) the barrier was 0.632 eV, in good agreement with similar efforts. Kürpick and Rahman [80] used embedded atom potentials with a cell of (10 × 10) atoms and 10 layers. They arrived at a barrier of 0.63 eVand a prefactor of 3.4 × 10−4 cm2/ sec with FBD [57] parameters, and 0.68 eV together with Do = 1.3 × 10−3 cm2/sec in the VC approximation. Atom exchange during diffusion was not considered. Later Kürpick [67] reported the same value of the diffusion barrier but with a prefactor of (3.7 ± 0.1)×10−3 cm2/sec. Merikoski et al. [81] used EMT potentials to calculate the barriers from the adiabatic energy surface for different diffusion mechanisms. They found an activation energy of 0.631 eV for atom hopping compared to 0.844 eV for atom exchange. In ab initio density-functional estimates, Feibelman [13] obtained an activation energy of 0.72 eV for self-diffusion by hopping on paramagnetic Ni(100). Embedded atom method potentials were again used by Mehl et al. [82] to calculate the hopping barrier. They worked with 100 atoms in each layer and 20 layers, finding a value of 0.63 eV, in excellent agreement with previous theoretical data. In 1999, Davydov [49] examined self-diffusion on Ni(100), relying on the cohesion based approach of Wills and Harrison [83]. For the activation energy he derived a value of 0.20 eV, which is clearly much too low. Haug et al. [84,85] looked at the influence of hydrogen on homo-epitaxial growth on Ni(100). They worked with 77 dynamically active atoms and estimated the barrier for self-diffusion by hopping, using EAM potentials, as 0.61 eV; exchange was not considered. Chang et al. [28,86] carried out ab initio density-functional calculations and discovered that atom exchange occurred over a barrier comparable to hopping: 0.78 eV for exchange compared to 0.82 for hopping. As had earlier been done by Perkins and DePristo [29], they also examined the effect of the number of atoms in the unit cell of the surface on the activation energy calculated. The results were plotted in Fig. 3.60, and it is clear that for atom hopping there is essentially no dependence on the number of surface atoms; this is not true for diffusion by atom exchange, where it is necessary to have a large region in which atoms can relax their positions. For Ni(100) the barrier for exchange changed from 1.39 eV to 0.78 eV, making exchange the more favorable mechanism than hopping. Agrawal et al. [26] depended on Lennard-Jones potentials in their estimates for the hopping barrier and found it to be 0.75 eV, with a prefactor of 2.5 × 10−3 cm2/sec. No consideration was given to the possibility of an exchange mechanism or many-body effects. In 2006, Kong and Lewis [69] reported the kinetic prefactor for self-diffusion on Ni(100) calculated with EAM potentials by summing the vibrations over all atoms. They found a value of 5.3 × 10−3 cm2/sec at 300 K, and 5.2 × 10−3 cm2/sec at 600 K. From molecular dynamics simulations using RGL interactions, Kim et al. [70] derived a barrier of 0.376 eV for atom hopping and a much bigger value of 1.304 eV for exchange. The value for hopping is surprisingly low, and not in agreement with either experiment or previous theoretical estimates.

5.6 Nickel: Ni(100)

281

It appears that independent of the technique used, calculated estimates for the selfdiffusion barrier on Ni(100) are reasonably close together and to experiments, but the estimates are not very satisfactory – they leave uncertain the mechanism of diffusion, which in experiments was established to be atom exchange. Less than half of the theoretical contributions have actually made calculations of the energetics for exchange. These results have a large spread, ranging from a barrier of 0.47 eV to 1.304 eV. The spread may be tied to the effect noted by Perkins and DePristo [29], as well as Chang et al. [86], that a considerable number of adjustable atoms must be included in the surface to accommodate strain. Values for the hopping energy of nickel on Ni(100), excluding work with pair interactions and the cohesive approximation of Davydov [49], are in much better shape; they are usually fairly close together.

5.6.2

Hetero-diffusion There have been a few attempts to understand diffusion of foreign species on nickel surfaces. However, only three experiments were attempted to look at hetero-diffusion. The first, in 1996, was done by Müller et al. [87] in Kern’s laboratory in Lausanne. They relied on nucleation theory to study the deposition of copper on the (100) plane of nickel, and utilized the scanning tunneling microscope as a tool to characterize the details of surface structure. Their results, obtained at a coverage of 0.1 monolayer, are shown in Fig. 5.17. For a critical nucleus size of one, implying stable dimers, they derived a diffusion barrier of 0.351 ± 0.017 eV and an attempt frequency ν0 = 4 × 1011 ± 0.3 sec−1. In the temperature range studied here, Müller et al. concluded that diffusion did not occur by atom exchange on the surface but rather by hopping. Diffusion of platinum atoms was examined on nickel surfaces by Kellogg [72] using the field ion microscope. From the disappearance of adatoms from the (100) and the observation of atoms in the surface layer resistant to field evaporation, he concluded that diffusion occurred by atom exchange. Assuming a prefactor of 1 × 10−3 cm2/sec, Kellogg estimated an activation energy of 0.69 eV. In 1993 Krausch et al. [88] reported data from perturbed γ–γ angular correlation spectroscopy (PACS) which used 111In as a probe. They deposited probe material in the range 10−4 ML on the surface at 77 K and monitored changes in frequency as the probe scanned over the deposit. For diffusion of In on Ni(100) they reported an activation energy of 0.31 ± 0.03 eV. No information about the mechanism was deduced from this study. There were also a few theoretical investigations of hetero-diffusion on Ni(100). The first was done by Sanders and DePristo in 1992 [76] using MD/MC-CEM. Activation energies and frequency prefactors were: for copper 0.62 eV and 5.2 × 1012 sec−1; for rhodium 0.67 eV and 4.5 × 1012 sec−1; for palladium 0.55 eV and 4.2 × 1012 sec−1; for silver 0.42 eVand 10 × 1012 sec−1; for platinum 0.55 eVand 3.7 × 1012 sec−1; and for gold 0.48 eV and 2.9 × 1012 sec−1. It should be noted that the activation energy found here for Cu diffusion is much larger than the value reported in experiments by Müller et al. [87], and the study considered only hopping as a possible mechanism of movement. The next attempt to establish the energetics, this time both for hopping and exchange, was made by Perkins and DePristo [89] who also looked at hetero-diffusion with 226

282

Diffusion on two-dimensional surfaces

T = 215 K

400

250

Temperature T(K) 160 125

nx (per lattice site)

1 post-nucleation 10

i=1

10–3

102

10–4

103 i=3

10–5 2 (a)

104 4

6

8

Mean Island Size (atoms)

10–2

100

T = 280 K

T = 345 K

10

1000/T (K–1)

(b) Fig. 5.17

300 Å

Deposition of Cu on Ni(100) plane at 0.1 ML coverage and a deposition rate of 1.34 × 10−3 ML/sec. (a) Arrhenius plot of saturation island density against reciprocal temperature. (b) STM images of copper growth at different temperatures (after Müller et al. [87]).

movable atoms in the first layer. They obtained the following diffusion barriers from MD/MC-CEM calculations: for hopping of rhodium 0.65 eV, for exchange of a rhodium atom with Ni 0.85 eV; for hopping of palladium atom 0.64 eV, for exchange of palladium 1.10 eV; for hopping of copper 0.45 eV while for exchange of Cu 1.23 eV; for hopping of silver 0.44 eV, for exchange 1.47 eV. The frequency prefactors, expressed as multiples of 1012 sec−1, were: for Rh hopping 8.28, for Rh exchange 9.37; for Pd hopping 9.77, for Pd exchange 11.93; for Cu hopping 4.15, for Cu exchange 3.76; for Ag hopping 7.80, for Ag exchange 4.45. The characteristics obtained for diffusion of copper on Ni(100) are in significantly better agreement with the experimental results of Müller et al. [87] than earlier estimates by Sanders and DePristo [76]. Unfortunately there are no experimental values with which to compare the rest of the data. One thing worth noting is that Perkins and DePristo [89] worked with a bigger cell, which should give a better prediction of the barrier for exchange. Kim et al. [70] estimated the following barriers from molecular dynamics simulations with RGL interactions: Cu 0.407 eV, Pd 0.492 eV, Ag 0.347 eV, Pt 0.706 eV and Au 0.489 eV. These values are always not in good agreement with the work of Perkins and DePristo [89], but are quite close to Kellogg’s [72] findings for platinum. Values obtained for copper are close to the experimental results of Müller et al. [87]. It must be noted that hetero-diffusion has not been carefully examined in theoretical calculations. Based on Perkins and DePristo’s [89] data, it seems that hopping is a more

283

5.7 Copper: Cu(100)

likely mechanism than exchange, but experimental investigations in this area are less than poor. For platinum on Ni(100) exchange is certain [72], but the estimates of Kim et al. [70], which gave a similar barrier, were for atom hops, not exchange.

5.7

Copper: Cu(100)

5.7.1

Self-diffusion: experimental work Diffusion measurements, which may be quite far from values for single atom movement on the copper (100) plane, started in 1987 and were done by de Miguel et al. [90]. They carried out thermal energy He atom scattering experiments on copper deposits of at least 0.5 monolayers evaporated onto the (100) plane of copper, and determined the mean terrace size over a range of temperatures. From this they were able to deduce an activation energy for diffusion of 0.40 eV, with a prefactor of Do = 1.4 × 10−4 cm2/sec. De Miguel et al. [91] also gave alternative procedures for deriving the diffusion characteristics: from the stationary intensities at different temperatures, where the scattered intensity no longer changes, and also from the temperature at which no oscillations are found in the specular intensity. Their results are plotted in Fig. 5.18 and give an activation energy of 0.48 eV and Do = 2.3 × 10−3 cm2/sec, in reasonable agreement with the previous results. Somewhat later, in 1992, Ernst et al. [92] examined the nucleation of copper on Cu(100) using He-atom beam scattering as a probe. From measurements of the mean separation between islands as a function of temperature at a coverage of 0.5 monolayers, they arrived at a much lower estimate for the diffusion barrier of 0.28 ± 0.06 eV, with a

Temperature T(K)

Diffusivity D(cm2/sec)

10–7

450

400

350

300

ED = 0.48 eV

10–8

Do = 2.3 × 10–3 cm2/sec

10–9 10–10 10–11 10–12

2.5

3.0

3.5

1000/T (K–1) Fig. 5.18

Arrhenius plot for self-diffusion of Cu on Cu(100) derived from thermal energy He atom scattering. Open circles: from stationary intensities. Open triangles: from temperature above which no oscillations are seen (after de Miguel et al. [91]).

284

Diffusion on two-dimensional surfaces

Yield (arbitrary units)

600

450

300

150

0 50 Fig. 5.19

100 150 Temperature T (K)

200

Plots for increasing and decreasing temperature dependence of copper adatom yield under ion beam irradiation of Cu(100), giving 145 K as temperature at which adatoms become mobile (after Breeman and Boerma [94]).

prefactor  10−5 cm2/sec. On reanalyzing this data with a different scaling factor, Brune et al. [93] obtained a value of 0.34 ± 0.07 eV for the activation energy of diffusion. The same year, Breeman and Boerma [94] produced copper adatoms on Cu(100) terraces, with a small coverage of indium for calibration purposes, by bombarding the crystal with ions (LEIS). The surface mobility was deduced from the number of atoms appearing after irradiation. This of course also created vacancies on the surface, which can influence the mobility, but it was assumed that all vacancies were eliminated at lattice steps. The yield of copper at different temperatures was measured to determine the conditions for the start of surface diffusion, as shown in Fig. 5.19. Assuming a frequency prefactor ν0 = 1 × 1013 sec−1, they arrived at a value of 0.39 ± 0.06 eV for the diffusion barrier, quite close to what de Miguel et al. [90] found. In 1995, Dürr et al. [95] also looked at the diffusion of copper on Cu(100), using high resolution LEED. From measurements of the island separation, shown in Fig. 5.20 at a coverage of 0.3 monolayers and for temperatures from 180 to 260 K, they arrived at an activation energy of 0.36 ± 0.03 eV relying on mean-field nucleation theory. At higher temperatures detachment of atoms from islands is possible. It must be noted that these results were obtained at a rather high coverage where interactions are likely to play a role, and that the technique used is not sensitive enough to control the state of the surface, so results for single atom movement are somewhat uncertain. In 1999 Laurens et al. [96] looked at the Cu(17,1,1) plane with PACS techniques; this plane consists mostly of (100) terraces. From the onset temperature for self-diffusion of Cu atoms they derived an activation energy on Cu(100) as 0.36 eV, in agreement with previous results, which are all reasonably close together.

285

5.7 Copper: Cu(100)

Temperature T (K) 280 260 240 220 200

Ff = 3.21 × 10–4 ML/sec Θ = 0.3 ML

Island separation (Å)

200

100 ED = 0.36±0.3 eV

4.0

Fig. 5.20

180

4.5 5.0 1000/T (K–1)

5.5

Arrhenius plot of Cu island separation on Cu(100) as a function of reciprocal temperature (after Dürr et al. [95]).

None of the above data are really ideal for drawing conclusions about single atom diffusion, since coverages are much too high and interactions may influence movement on the surface. Unfortunately, the mechanism of motion also did not emerge from these studies.

5.7.2

Self-diffusion: theoretical work Many theoretical estimates of the self-diffusion parameters for Cu(100) have been made, starting very early, in 1968, with the work of Wynblatt and Gjostein [97]; they did calculations relying on Morse potentials to arrive at a self-diffusion barrier of 0.248 eV. Also using Morse potentials, Flahive and Graham [47] found a self-diffusion barrier of 0.27 eV, as already shown in Fig. 4.9. The first many-body calculations on Cu(100) were carried out by Liu et al. [4], who derived a barrier to hopping of 0.38 eV, evaluated using AFW potentials, with a prefactor of 1.2 × 10−3 cm2/sec; with VC potentials they arrived at a barrier of 0.53 eV and a prefactor of 5.2 × 10−3 cm2/sec. The first of these estimates is in good agreement with experiments. For exchange on this plane the barrier was much bigger, 0.72 eV (AFW) and 0.79 eV (VC) with a prefactor Do = 2 × 10−2 cm2/sec. Hansen et al. [98] depended upon effective medium theory potentials to obtain a rather low barrier of ~ 0.22 eV. This, it should be noted, was for diffusion by atom exchange; the hopping barrier proved much higher, 0.43 eV. Hansen et al. [99] continued their work in 1993 and evaluated the free energy of activation, both by atom hopping and by exchange. Free energies were evaluated for a range of temperatures, but we confine our attention to ~ 300 K. Here the barrier to atom jumps was 0.398 eV, and only ~ 0.200 eV for diffusion by atom exchange, as shown in Fig. 5.21. The prefactors for the two processes were 4.7 × 10−4 cm2/sec and 8.5 × 10−4 cm2/sec.

286

Diffusion on two-dimensional surfaces

0.46 0.29 Activation energy (eV)

Activation energy (eV)

0.27 0.25 0.23 0.21 0.19 (a) 0.17 0.15

Fig. 5.21

Total energy Free energy

Total energy Free energy

0.44

0.42

0.40

(b)

Exchange diffusion 0

100 200 300 Temperature T (K)

Bridge diffusion 400

0.38

0

100 200 300 Temperature T (K)

400

Temperature dependence of the activation energy and free energy of adatom self-diffusion on Cu(100) by (a) atom exchange and (b) atom hopping; calculations by effective medium theory (after Hansen et al. [99]).

Transition-state estimates of the rate of self-diffusion were done a year later by Sanders and DePristo [76] using MD/MC-CEM theory to arrive at interaction energies. They found a barrier of 0.66 eV with a frequency prefactor of 5.5 × 1012 sec−1. In this work only atom hopping was examined; the possibility of atom exchange in diffusion was not considered. Perkins and DePristo [77] continued their previous study using the same method with 13 atoms active at the surface. They found a barrier to hopping of 0.52 eV, with a much higher value of 1.1 eV for exchange. Using EAM potentials in the VC approximation they arrived at much the same barrier, 0.53 eV, and 0.79 eV for atom exchange. In the same paper they also made estimates with CEM potentials and found an activation energy of 0.47 eV for hopping compared to 0.43 eV for atom replacement. From the CEM method they concluded that replacement was the preferred mechanism in self-diffusion on Cu(100) and that the kinetic energy correlation term was crucial in the correct description of the exchange mechanism. However in 1995, they [29] reanalyzed their previous results with an increase in the number of movable atoms in the outer layer from 13 to 226. For hopping the barrier changed only little, from 0.52 eV to 0.51 eV using MD/MC-CEM, and from 0.47 eV to 0.46 eV with CEM. However, in diffusion by atom exchange the activation energy was altered significantly, from 1.1 eV to 0.77 eV with MD/MC-CEM and from 0.43 eV to 0.18 eV using CEM, seemingly making atom exchange the favored mechanism. Their data are still not conclusive since CEM gave a lower value for exchange, while MD/MC-CEM favored hopping. Molecular dynamics simulations using embedded atom potentials were carried out by Black and Tian [8]. Their work was done for different diffusion temperatures. At 600 K they found atom exchange processes, but most interesting were simulations at 900 K. An adatom entered the substrate, and a lattice atom not adjacent to the entering atom came out of the substrate. The strain introduced by the entering atom was believed to propagate

287

5.7 Copper: Cu(100)

–7.0 Cu on Cu(100)

In D

–7.5

Morse potential SC potential

–8.0 –8.5 –9.0 250

300

350

400

450

500

1/T Fig. 5.22

Dependence of Cu adatom diffusivity on Cu(100) plane upon reciprocal temperature. Calculations by Sutton–Chen potentials yielded a diffusion barrier of 0.243 eVand a prefactor of 8.62 × 10−3 cm2/sec. Using Morse potentials, the values were 0.249 eV and a prefactor of 7.80 × 10−3 cm2/sec (after Shiang [102]).

along a row of substrate copper atoms. Eventually an atom emerged from the substrate at some distance from the original point of entry. Unfortunately they did not derive the energetics for this process, but this study showed that the exchange mechanism should contribute to movement on this surface, even if most of theoretical studies concluded that exchange was more demanding than hopping. Embedded atom estimates were made by Tian and Rahman [100], who reported a hopping barrier of 0.49 eV; only exchange with edge atoms was investigated, and surprisingly this turned out to be comparable to terrace hopping. Many-body potentials introduced by Sutton and Chen [101] were used by Shiang [102] to do molecular dynamics simulations shown in Fig. 5.22. He calculated a barrier of 0.243 eVand a prefactor of 8.62 × 10−3 cm2/sec. Using Morse potentials, without concern for many-body effects, he arrived at almost the same barrier of 0.249 eVand a prefactor of 7.80 × 10−3 cm2/sec. It is possible that for Cu(100) many-body effects are not so important as for Ni. Stoltze [61] estimated an activation energy of 0.425 eV for jumps of copper atoms on Cu(100) using effective medium potentials. Liu [103] depended on embedded atom interactions in arriving at an energy barrier of 0.45 eV for hopping. Lee et al. [104] performed ab initio calculations to find a barrier of 0.69 eV for hopping of copper adatoms, and a barrier of 0.97 eV for atom exchange. Karimi et al. [82,105, 106] made estimates for the diffusion of copper atoms relying on EAM potentials. For hopping they obtained a barrier of 0.48 eV, compared with 0.80 eV for diffusion by atom exchange. Effective medium theory was employed by Merikoski and Ala-Nissila [107] to arrive at a barrier of 0.399 eV for hopping. Subsequently Merikoski et al. [81] also worked out an activation energy of 0.996 eV for exchange. In 1996, Shi et al. [79] used EAM potentials to establish an activation energy of 0.503 eV for hopping; atom exchange was not considered. Diffusion of copper on Cu(100) was examined with a smart Monte Carlo technique by Kumar et al. [108]. Using MD/MC-CEM potentials, they arrived at an activation energy of 0.52 ± 0.04 eV and a prefactor of 1.6 ± 0.3 × 10−3 cm2/sec and showed that this value was independent of the time step used.

288

Diffusion on two-dimensional surfaces

Using an RGL type potential [109,110], Evangelakis and Papanicolaou [111] resorted to molecular dynamics simulations to find a barrier of 0.43 ± 0.02 eV, with Do = 3.4 × 10−3 ± 0.2 cm2/sec for hopping, compared with 0.70 ± 0.04 eVand Do = 42 × 10−3 ± 0.3 cm2/ sec for atom exchange. An additional important development was that they also arrived at the characteristics of long transitions: for double jumps a barrier of 0.71 ± 0.05 eV and a prefactor of 38 × 10−3 ± 0.5 cm2/sec; double exchange 0.70 ± 0.06 eV with 45 × 10−3 ± 0.6 cm2/sec; triple exchange 0.82 ± 0.08 eV and 104 × 10−3 ± 1.0 cm2/sec; as well as quadruple exchange 0.75 ± 0.12 eV and 86 × 10−3 ± 0.9 cm2/sec. Their study confirmed the earlier work of Black and Tian [8] as well as Cohen [7]. An example of such a long-range transition seen at 950 K is shown in Fig. 3.64. It was possible to gain this detailed information as data were obtained at high temperatures ranging from 700 K to 1100 K. Boisvert and Lewis [112] worked with the ab initio density-functional method using the generalized gradient approximation. For atom jumps they found a barrier of 0.52 ± 0.05 eV, and for exchange 0.96 ± 0.1 eV. However, their surface cell was only 3 × 3, and they used a block of four layers, which could have affected the energy estimates for atomic exchange. With molecular dynamics using EAM interactions, they found a barrier of 0.49 ± 0.01 eV for hopping with a prefactor of 20(× e±0.2) × 1012 sec−1 and 0.70 ± 0.04 eVand a prefactor of 437(× e±0.7) × 1012 sec−1 for atom exchange. A year later thermodynamic integration was used by Boisvert et al. [113] to evaluate transition-state barriers. They obtained almost the þ4:03 × 1012 sec−1 for atom same value of 0.51 ± 0.02 eV with a frequency prefactor of 18:2
3:29 hopping. For atom exchange the barrier was slightly higher at 0.74 ± 0.02 eV, with a þ5:47 × 1014 sec−1. frequency prefactor 6:65
3:00 Relying on EAM potentials and transition-state theory, Kürpick and Rahman [80,114] evaluated the diffusion characteristics of copper on Cu(100) at 600 K using both FBD [57] and VC [6] parameters. For the former the diffusion barrier amounted to 0.51 eV and the prefactor was 3.2 × 10−4 cm2/sec; with the VC approximation a barrier of 0.53 eV was obtained with a prefactor of 3.3 × 10−4 cm2/sec. The barriers for the expanded system at 600 K were very similar, 0.53 eV with a prefactor of 3.8 × 10−4cm2/sec from FBD parameters and 0.55 eV and 4.7 × 10−4cm2/sec from VC. Two years later Kürpick and Rahman [115] resorted to the same methods to arrive at the same values of the activation energies but with different prefactors. With FBD parameters the prefactor was 8.7 × 10−4 cm2/sec, using VC potentials it turned out to be 9.0 × 10−4cm2/sec. Li et al. [11] relied on the discrete variational method with the local-density-functional approximation to calculate activation energies for different types of transitions. For atom hopping along the barrier proved to be 0.93 eV, for atom exchange it was much higher, 2.81 eV, and for hops over atom tops along it was 1.25 eV. Hopping was clearly the lowest energy process, but the barriers calculated here are much higher than in other studies. Trushin et al. [116,117] used EAM potentials to find an activation energy of 0.49 eV for atom jumps and 0.69 eV for atom exchange. Xie [118] used the embedded atom method to calculate interactions for molecular dynamics simulations and he arrived at a barrier of 0.56 eV for hopping over an unrelaxed surface, and claimed that relaxation changed the barrier by less than 0.1 eV; no estimates were made for atom exchange. Davydov [49] in 1999 arrived at a barrier of 0.01 eV for atom hopping using his cohesion approximation. This value is entirely out of line compared

5.7 Copper: Cu(100)

289

to other estimates. Zhuang and Liu [119] used RGL potentials to look at exchange and jump processes. For simple exchange they obtained a barrier of 0.70 ± 0.04 eV, from static calculations the value was 0.85 eV. However, the barrier to simple atom hopping proved to be much lower, 0.44 eV from an Arrhenius plot. Adams et al. [120,121] used EAM potentials in the AFW approximation to evaluate the activation energy. Their work yielded a barrier of 0.50 eV for atom hopping and 0.75 eV for atom exchange, with the former process clearly preferred. They also showed that bond counting did not properly account for surface diffusion effects. Four years later, Wang et al. [122] did molecular statics calculations using embedded atom potentials and arrived at an activation energy of 0.48 eV for adatom hopping. The exchange mechanism led to a higher value of 0.69 eV. Kürpick [67] used a slightly different approach than previously; with RGL potentials and transition-state calculations she found a barrier of 0.44 eV and a prefactor of (2.5 ± 0.1) × 10−3 cm2/sec for hopping over the surface. In the same year, Fordell et al. [21] carried out ab initio calculations and in the LDA approximation they arrived at a barrier to atom hopping of 0.67 eV, and for GGA the activation energy was 0.55 eV. Exchange barriers proved to be much higher, 0.94 eV for LDA and 0.81 eV for GGA. Lennard-Jones potentials were used by Agrawal et al. [26] to calculate an activation energy of 0.59 eV with a prefactor of 2.3 × 10−3 cm2/sec for copper adatoms hopping, but completely ignored all many-body effects. However Liu et al. [4] already showed in 1991 that many-body effects do not play a crucial role in the case of copper. Pentcheva [123] relied on density-functional theory to look at the diffusion of copper adatoms and she found a barrier of 0.49 eV for jumps of copper, and a much larger barrier of 1.02 eV for diffusion by atom exchange. Activation energies were evaluated by Chang and Wei [28] using the VASP simulation routines [22–24]. For atom hopping they arrived at a barrier of 0.64 eV, but found a much higher value of 0.94 eV for atom exchange. Effective medium estimates were used by Jahma et al. [124] to look at diffusion and submonolayer growth by kinetic Monte Carlo simulations. For self-diffusion they briefly mentioned a barrier of ~0.4 eV, and assumed hopping as the leading mechanism. Another calculation of atom hopping with embedded atom potentials was made by Yildrim et al. [125] with EAM potentials using the local-density approximation. They found an activation energy of 0.505 eV for jumps, and a prefactor at 300 K of 7.29 × 10−4 cm2/sec; at 600 K this changed very little to 7.45 × 10−4 cm2/sec. This local-density approximation has been challenged by Kong and Lewis [126], who used the same EAM potentials but calculated the prefactor by summing over all frequencies. They obtained precisely the same activation energy for diffusion. For the prefactor to the diffusivity, however, they obtained a different value, 5.3 × 10−3 cm2/sec at 300 K, which changed to 5.8 × 10−3 cm2/sec at 600 K, and they concluded that the local-density approximation was not in general valid. The activation energy was calculated by Müller and Ibach [127] using the localdensity approximation; they arrived at a value of 0.67 eV for the barrier to hopping, compared to 1.02 eV for atom exchange. For self-diffusion by atom jumps Kim et al. [70] estimated a barrier of 0.477 eV, and for exchange 0.708 eV, both from molecular dynamics simulations relying on RGL potentials. These results are in reasonable accord with much of the prior calculations, but higher than most experimental values.

290

Diffusion on two-dimensional surfaces

The characteristics for self-diffusion on Cu(100) are not in a very good state. Four experiments have been done, and an average barrier of 0.37 ± 0.03 eV emerges from these. The results are reasonably close together, but it must be noted that all the studies have been based on indirect deductions, without any direct observations of atom motion. Theoretical estimates of atom hopping are in worse shape. There are roughly thirty of these and the more reasonable ones for hopping range from 0.243 eV to 0.69 eV, with an average value of 0.49 ± 0.06 eV. In more than a dozen of these evaluations, estimates were made of atom motion by exchange with the lattice; in these, reasonable energies ranged from 0.18 eV to 1.02 eV, with a mean value of 0.69 ± 0.30. Early theoretical studies indicated exchange as the primary mechanism of movement, the latest definitely favor hopping. There is not enough experimental information to verify which prediction is correct. At the moment it is therefore not clear which mechanism is primarily responsible for diffusion, and it appears that further work, mostly experimental, would be desirable.

5.7.3

Hetero-diffusion Some experiments on the diffusion of indium on Cu(100) were done by Breeman and Boerma [128] using low energy neon ion scattering. They looked at the fraction of In atoms deposited on different sites at various deposition temperatures. With 0.013 ± 0.002 monolayers of indium on the surface, they were able to determine the temperature at which indium became mobile as being between 77 K and 88 K. Assuming an attempt frequency of 1013 sec−1, they arrived at an activation energy of 0.24 ± 0.03 eV for atom hopping, and were unable to account for their observations assuming diffusion occurred by atom exchange. In 1993 Krausch et al. [88] reported diffusion data obtained using perturbed γ−γ angular correlation spectroscopy (PACS). This technique used 111In in the range 10−4 ML as a probe. Indium was deposited on the surface at 77 K, and atom movement as well as changes in frequency were monitored. The activation energy for movement of In on Cu(100) was reported as 0.28 ± 0.05 eV. Diffusion into the bulk required a huge energy of 2.05 ± 0.05 eV. Movement of 111In on Cu(17,1,1), quite close to the (100) plane, was also investigated by Rosu et al. [129] in 2001, again using the PACS technique. Coverage of indium was less than 10−3 ML and an activation energy of 0.20 ± 0.03 eV was found. Diffusion of indium atoms was also studied by Van Siclen [130] using EAM interactions. For diffusion, the energy barrier was estimated as 0.32 eV. Diffusion by atom exchange involved much more energy, requiring 1.01 eV, in agreement with the experimental findings of Breeman and Boerma [128], but their value of the hopping energy was slightly higher. Cohen et al. [131] looked at the spreading of a two-dimensional submonolayer of Pb deposited on Cu(100) as a function of coverage. Measurements were done by Rutherford backscattering. From the temperature dependence of the diffusivity at low coverages they deduced a barrier of 0.68 ± 0.1 eV for isolated jumps of Pb atoms. Graham et al. [132] also examined the diffusion of sodium atoms on Cu(100) using high resolution scattering of helium atoms. From measurements of the peak width as a function of the temperature they were able to come up with the Arrhenius plot in

291

5.7 Copper: Cu(100)

400

700

Temperature T(K) 300 250 200

150

Simulated Result

500

Fit to Experiment

400

Peak Width Γε (μeV)

300 200 150 100 70 50 40

Cu(100) EHe = 11.2 meV ΔK = 0.86 Å–1

30 20

Fig. 5.23

2

3

4 5 1000/T (K–1)

6

7

Dependence of quasi-elastic peak width upon reciprocal temperature in He scattering from Na atoms at 0.028 ML coverage on Cu(001). Best fit gives activation energy of 0.0525 ± 0.0009 eV. Predicted slope 0.065 ± 0.006 eV (after Graham et al. [132]).

Fig. 5.23, yielding an activation energy of 0.0525 ± 0.0009 eV. The dashed line indicates the curve predicted by molecular dynamics simulations for temperatures T 280 K with a barrier of 0.065 ± 0.006. These results were obtained for a sodium coverage of 0.028, which may yield the behavior of individual adatoms. The data are in excellent agreement with earlier measurements obtained in the same group by Ellis and Toennies [133]. This yielded an activation energy of 0.051 ± 0.006 eV and a frequency prefactor of (0.53 ± 0.2) × 1012 sec−1 at a higher sodium concentration of 0.1 layer, as shown in Fig. 5.24. In 1997 Graham et al. [134] made an attempt to find the diffusivity D in the limit when the wave vector ΔK→0. They claimed that the experimental sensitivity in this region was too low for direct extraction of the diffusivity, and instead extrapolated via the calculated quasi-elastic width to the limit ΔK→0 and used the zero frequency Laplace transform of the velocity autocorrelation function of Na atoms. By this procedure they found Do = (6.3 ± 0.6)×10−3cm2/sec and ED= 0.075 eV. It was claimed that the difference between the earlier value of the barrier and the present one arose because the quasi-elastic peak had contributions from diffusive motion and T-mode vibrations at finite ΔK. The real barrier corresponded to 0.075 eV, not 0.052 eV as suggested previously. They also predicted a value of 0.065 ± 0.006 eV for the effective barrier for diffusion. In 1999, Cucchetti and Ying [135] carried out molecular dynamics simulations for the movement of Na on the Cu(100) surface using their own model Hamiltonian. Their

292

Diffusion on two-dimensional surfaces

500

Temperature T (K) 333 250 200

167

ED = 0.051±0.006 eV ν0 = 0.53 × 1012 sec–1

Energy Width ΔEε (μ eV)

500 400 300 200

100 50 2

Fig. 5.24

3

4 1000/T (K–1)

5

6

Arrhenius plot of He peak width in scattering from Na atoms on Cu(001) plane, yielding an activation energy of 0.051 eV (after Ellis and Toennies [133]).

Barrier for diffusion (meV)

95 90 85 80 75 70 65

Fig. 5.25

0

0.05

0.1 Coverage

0.15

0.2

Activation energy for tracer diffusion of Na on Cu(001) surface calculated for different coverages (after Cucchetti et al. [135]).

results for tracer diffusion are shown in Fig. 5.25. The value of the barrier at a very low coverage was around 0.077 eV, very close to the findings of Graham et al. [134]. Recently the system was investigated again with 3He spin echo measurements by Alexandrowicz et al. [136] at lower coverages of 0.02 ML and 0.08 ML; the lower coverage can probably be associated with interaction-free movement. Their data are shown in Fig. 5.26. The activation energy deduced at the lower coverage was 0.058 ± 0.007 eV, rather close to the earlier estimates of Graham et al. [132], and diffusion was by atom hopping. What is more important is that they came out with a value of 0.042 ± 0.005 eV for motion perpendicular to the surface, although there was no previous evidence of atoms getting away from the surface during jumps, and this value was obtained for a higher coverage

293

5.7 Copper: Cu(100)

Inverse lifetime (nsec–1)

60 45

ED = 0.042 ±0.005 eV

30

15

ED = 0.058 ±0.007 eV

0.0055 0.0065 1/T (K–1) Fig. 5.26

0.0075

Arrhenius plots for the inverse lifetime of Na on Cu(001), obtained in 3He spin echo experiments. Inverted triangles: 2D motion, coverage 0.02 ML; circles: perpendicular movement, coverage 0.08 ML (after Alexandrowicz et al. [136]).

of 0.08 ML. The same group continued investigations of Cs on Cu(100) at a coverage of 0.014 and 0.044 ML [137]. From their experiments they deduced a value of 0.31 ± 0.02 eV for the diffusion barrier at a coverage of 0.044 ML; from Langevin molecular dynamic simulations they obtained a value of 0.24 ± 0.04 eV, and observed multiple jumps during movement of Cs over the surface. Sanders and DePristo [76] studied hetero-diffusion in some detail relying on MD/ MC-CEM interactions. They first did molecular dynamics simulations and compared the results with estimates using transition-state theory (TST) and found good agreement. With this established, they worked with hetero-atoms on Cu(100), and arrived at the following characteristics from TST: nickel diffusion barrier of 0.64 eV and a frequency prefactor 5.2 × 1012 sec−1; rhodium 0.71 eV and 4.6 × 1012 sec−1; palladium 0.58 eV and 4.6 × 1012 sec−1; silver 0.36 eV and 6.9 × 1012 sec−1; platinum 0.61 eV and 4.1 × 1012 sec−1; gold 0.56 eV and 3.1 × 1012 sec−1. No consideration was given to the possibility that atom exchange might contribute to diffusion. However, the frequency prefactors all seem to lie in the right range, ~ 1012 sec−1. Further work was done by Perkins and DePristo [89], who still continued with MD/ MC-CEM and transition-state theory, but now also worked out what happens when atom exchange contributes. Their results for a surface consisting of 226 movable atoms were as follows for the hopping characteristics: rhodium 0.72 eV and 12.43 × 1012 sec−1; nickel 0.64 eV and 9.44 × 1012 sec−1; palladium 0.73 eV and 17.57 × 1012 sec−1; silver 0.48 eV and 9.07 × 1012 sec−1. For diffusion by atom exchange their results were rather different: rhodium 0.42 eV and 10.61 × 1012 sec−1; nickel 0.57 eV and 9.00 × 1012 sec−1; palladium 0.59 eVand 15.58 × 1012 sec−1; silver 0.91 and 7.5 × 1012 sec−1. It appears that for Rh, Ni, and Pd, diffusion by exchange of atoms is really the favored process, but not for Ag. Values for Rh and Ni hopping agreed very well with the previous findings of Sanders and DePristo [76].

294

Diffusion on two-dimensional surfaces

Gold atoms on Cu(100) were examined by Evangelakis et al. [138] with RGL potentials; from the Arrhenius plot in Fig. 5.27 they found an activation energy of 0.58 ± 0.02 eV for atom exchange. The barrier estimated from energy minimization was 0.57 eV for exchange, and 0.64 eV for atom hopping. Pentcheva [123] did density-functional theory calculations in the WIEN97 implementation for diffusion of both copper and also cobalt atoms on Cu(100). For hopping of cobalt the barrier proved to be 0.61 eV, and 1.00 eV for atom exchange, as is illustrated in Fig. 5.28. However, for non-magnetic cobalt adatoms the hopping barrier proved to be much higher, 0.92 eV. Both copper and cobalt appear to diffuse by atoms hopping over the surface. First-principles calculations with RGL potentials for movement of Co on Cu(100) as well as on Co36 strained islands were performed by Stepanyuk et al. [139]. Temperature T(K) 900 1000

Frequency ν (1012 sec–1)

1100

800

Exchange ED = 0.58±0.02 eV

0.1

0.01 10

11

12

13

14

15

1/kT (eV–1) Fig. 5.27

Arrhenius plot for diffusion of Au atom on Cu(100) plane evaluated using RGL potentials (after Evangelakis et al. [138]).

1.5 Energy E(eV)

TS 1.0 0.5

IS

0.0 FS –0.5 –1.0 1 2 3 0 Reaction coordinate (bohr)

Fig. 5.28

4

Potential diagram for exchange of Co adatom with lattice atom on Cu(100), calculated by density-functional theory. Adatom hopping is preferred (after Pentcheva [123]).

295

5.8 Copper: Cu(111)

Movement of Co was preferred by hopping, with an energy of 0.66 eV, while exchange required 0.86 eV. Diffusion on Co36 islands required 0.58 eV by hopping and 0.90 eV by exchange. Movement along the islands was, however, a very low energy process with a barrier of 0.20 eV. What is worth noting is that the misfit between Co and Cu is about 2%, so a Co36 island is under compressive strain. The amount of such strain decreases with the size of the island, but is present for Co100 islands as well. It is interesting that such an island creates a depression in the Cu(100) surface, so not only the Co36 island, but also the substrate underneath is under strain. Such depression becomes smaller for an island under 200 atoms. Kim et al. [70] also made barrier estimates for hetero-atom diffusion using molecular dynamics simulations based on RGL interactions. For Ni on Cu(100) the barrier to atom hopping was 0.439 eV, for Pd 0.566 eV, for Ag 0.393 eV, for Pt 0.874 eV, and for Au 0.554 eV; results rather different from what was obtained in previous attempts. It appears there is quite a lot of information available about hetero-diffusion on Cu(100), but existing data are difficult to compare and judge.

5.8

Copper: Cu(111)

5.8.1

Self-diffusion

Island number density (Å2)

The first experimental work which delivered the characteristics for diffusion on Cu(111) was done by Wulfhekel et al. [140] in 1996. They measured the saturation density of copper islands at a coverage of 0.1 monolayer using thermal energy helium scattering, and assumed that dimers were stable in the investigated range of temperatures. From the þ0:01 eV, data in Fig. 5.29 they arrived at an activation energy for self-diffusion of 0:03
0:005 but this may not be the value for single atom movement since the coverage was rather high. In 1994, Henzler et al. [141] analyzed spot profile LEED and at a coverage of 0.5 ML derived an activation energy of 0.064 eV from the island density, assuming a stable nucleus equal to one. For a Cu layer covered with 1 ML of Pb, movement required overcoming a barrier of 0.083 eV. It is unlikely, however, that these data are for the

10–5

ED = 0.03 eV

10–6 200 150

Fig. 5.29

100 Temperature T (K)

50

Saturation density of Cu islands on Cu(111) determined by thermal energy He scattering (after Wulfhekel et al. [140]).

296

Diffusion on two-dimensional surfaces

movement of single atoms not influenced by interactions, since the coverage was high. In 2000 Schlösser et al. [142] looked at vacancy island movement with a fast STM and modeled their findings with EMT potentials using Stoltze’s [61] program, to find for terrace self-diffusion a barrier of 0.057 eV. Excellent direct studies of atom jumps in self-diffusion on the Cu(111) plane have been carried out by two groups. In 2000, Repp et al. [143] reported STM observations of copper atoms on Cu(111), aimed at examining interactions between the adatoms. As part of that investigation, they also looked at the rate of atom jumps at a coverage of 0.003 ML. From the temperature dependence in the range from 9 to 21 K they arrived at a diffusion barrier of 0.037 ± 0.005 eV and a frequency prefactor of 5 × 1013 ± 1 sec−1. What is noteworthy here, besides the fact that the results were obtained directly from the measured rate of atom jumps and not from nucleation theory, is the small size of the diffusion barrier, which in the past has been connected with an unusually small prefactor [32]. The present results disprove such a connection. Two years later, Knorr et al. [144] published a more detailed study of the same system. From STM observations of isolated copper adatoms they found the rate of jumping at different temperatures around 13 K, as shown in Fig. 5.30, and arrived at a diffusion barrier of 0.040 ± 0.001 eV and a frequency prefactor ν0 = 1 × 1012 ± 0.5 sec−1. Diffusion was observed between only one type of adsorption site, presumably an fcc site. In 2003 Repp et al. [145] carried out further studies using the scanning tunneling microscope supplemented by ab initio calculations using the DACAPO code. They were able to establish an hcp-to-fcc hopping rate 75 times higher than for fcc-to-hcp, which established a lower limit of 0.004 eV for the energy difference between the two sites. From the difference in the temperatures for the start of diffusion out of the sites they came up with an upper limit of 0.008 eV, with binding at fcc sites the stronger. Calculations led

ED = 0.040±0.001 eV ν0 = 1 × 1012±0.5 sec–1

In[Γ (×10–12 sec–1)]

–5

–6

–7

–8 (a) 0.070

Fig. 5.30

(b) 0.074 1/T (K–1)

100 Å

0.078

Self-diffusion of Cu on Cu(111) measured with STM. (a) Arrhenius plot yields an energy barrier 0.040 ± 0.001 eV and a frequency prefactor ν0 = 1 × 1012 ± 0.5 sec−1. (b) Excerpt from sequence of STM measurements tracing diffusion at 13.5 K, 0.0014 ML (after Knorr et al. [144]).

297

5.8 Copper: Cu(111)

to a 0.006 eV stronger binding on fcc sites, as well as a diffusion barrier of 0.050 eV with a prefactor of 1 × 1012 sec−1. Agreement between the measurements is good, and lots of theoretical estimates have been made, even before any experiments had been done. Wynblatt and Gjostein [97] did energy calculations with Morse potentials and obtained a value between 0.023 eV and 0.032 eV for the barrier to self-diffusion. Also using Morse potentials, Flahive and Graham [47] calculated a barrier close to 0 eV, as in Fig. 4.9. The first estimates with a many-atom potential, EAM, were carried out by Liu et al. [4] in 1991. They found a barrier of 0.026 eV with AFW potentials and 0.044 eV when using VC. In these two cases the prefactors were 3 × 10−4 cm2/sec and 4.6 × 10−4 cm2/sec. Hansen et al. [98] in the same year made estimates with EMT potentials and arrived at a hopping barrier of ~ 0.12 eV. They also showed that exchange was unfavorable on this plane, requiring around 0.5 eV. The many-body potential of Sutton and Chen [101] was used by Shiang [102] to evaluate the activation energy from an Arrhenius plot, shown in Fig. 5.31. He found a barrier of 0.059 eV with a prefactor of 6.4 × 10−4 cm2/sec. Surprisingly enough the barrier obtained using Morse potentials, 0.064 eV with a prefactor of 5.24 × 10−4 cm2/sec, was much the same. In 1993, Kief and Egelhoff [15] evaluated the barrier for self-diffusion on Cu(111) as 0.14 eV by taking the results for Ir(111) and scaling according to the cohesive energies, but this value is much too high. Using the same sort of potentials as Hansen et al. [98] had used earlier, Stoltze [61] found a lower value, 0.053 eV. Karimi et al. [105] came up with a self-diffusion barrier of 0.028 eV using EAM, exchange cost 1.12 eV. Li and DePristo [62] arrived at a barrier of 0.039 eV with corrected effective medium theory. Kallinteris et al. [146], in 1996, did molecular dynamics simulations using the semiempirical RGL potential [110,147]. As shown in Fig. 5.32, they covered a range of temperatures, and found a diffusion barrier of 0.041 ± 0.002 eV as well as a prefactor

–8.0 Cu on Cu(111)

–8.1

Morse potential SC potential

–8.2

In D

–8.3 –8.4 –8.5 –8.6 –8.7

Morse: ED = 0.064 eV Do = 5.24 × 10–4 cm2/sec SC: ED = 0.059 eV Do = 6.4 × 10–4 cm2/sec

–8.8 250

300

350

400

450

500

1/T Fig. 5.31

Arrhenius plot for self-diffusion on Cu(111), calculated with Sutton–Chen and Morse potentials to yield a diffusion barrier of 0.059 eV (after Shiang [102]).

298

Diffusion on two-dimensional surfaces

ED = 0.087±0.008 eV Do = (7±1)x10–4cm2/sec

10–4

ED = 0.041±0.002 eV Do = (2±0.1) × 10–4cm2/sec

10–5 10–6 (a) 10–7

0

20

40

60

80

100 120 140 160

1/kT (eV–1) Fig. 5.32

10–3 Diffusivity D(cm2/sec)

Diffusivity D(cm2/sec)

10–3

ED = 0.087±0.008 eV Do = (7±1) × 10–4cm2/sec

10–4

(b) 10–5

10

15

20 1/kT (eV–1)

25

30

Self-diffusion of Cu adatoms on Cu(111) plane as a function of the reciprocal temperature, derived by molecular dynamics simulations with RGL potentials. (a) Overall temperature range. (b) Only high temperatures (after Kallinteris et al. [146]).

Do = (2 ± 0.1)×10−4 cm2/sec at lower temperatures from 80 K to 300 K. For direction the barrier was 0.043 ± 0.001 eV, perpendicular to the direction it was almost the same, 0.040 ± 0.001 eV. Agreement with experimental results is excellent. From their Arrhenius plot at higher temperatures, from 400 K to 900 K, they arrived at a barrier of 0.087 ± 0.008 eV and a prefactor Do = (7 ± 1) × 10−4 cm2/sec. In direction the energy barrier was 0.082 ± 0.007 eV, and in the perpendicular direction 0.10 ± 0.01 eV, so no anisotropy for diffusion was observed. At temperatures higher than 1000 K exchange started to be present with an energy of 1.19 eV. Using EAM interactions, Breeman et al. [148] calculated the activation energies for copper atoms hopping from fcc to hcp sites as 0.13 eV, and 0.11 eV for the reverse jumps. Mortensen et al. [63] estimated the barrier to self-diffusion as 0.16 ± 0.02 eV with the local-density approximation of density-functional theory and 0.12 ± 0.02 eV with the generalized gradient approximation. These estimates are rather too high. EAM potentials were also employed by Trushin et al. [116] in molecular statics estimates of the activation energy for self-diffusion on copper. On Cu(111), the barrier was 0.029 eV. Estimates of the barrier to self-diffusion were made in 1999 by Davydov [49] using his cohesion approximation, and he found a value of 0.05 eV. Although some of his estimates have come far from most calculations, in this instance agreement is quite reasonable with the experimental value of 0.04 eV [143,144]. Bogicevic et al. [149] carried out density-functional estimates with a cell 12 × 4 × 4 consisting of 192 atoms and obtained a barrier of 0.05 eV; they also pointed out that the barrier changed by a factor of two if the surface was not relaxed. With embedded atom interactions, Chang et al. [35] once more found the barrier to self-diffusion to be 0.05 eV. Relying on RGL potentials, Kürpick [67] estimated a diffusion barrier of 0.042 eV with a prefactor of 1.2 ± 0.1 × 10−4 cm2/sec, one of not too many theoretical studies which derived the prefactor as well. Larsson [150] studied island diffusion on Cu(111) with Monte Carlo simulations, relying on bond energies derived in the literature. For the activation energy of single copper adatom diffusion he obtained a barrier of 0.09 eV, much higher than the experimental value.

299

5.8 Copper: Cu(111)

Agrawal et al. [26] in 2002 used Lennard-Jones potentials to calculate a barrier of 0.11 eV and a prefactor of 0.14 × 10−3 cm2/sec for hopping. Just as for other systems, this estimate is much too high. In 2003 Wang et al. [151] looked at the influence of strain on the mechanism of movement on Cu(111), using MD simulations based on EAM potentials [152]. They worked at three temperatures, 650 K, 800 K, and 950 K, and demonstrated that the diffusivity decreased as a function of increasing uniaxial and biaxial stress, as shown in Fig. 3.74. On analyzing a trajectory they found both hopping as well as atom exchange – it turned out that stress can change the mechanism of movement. In the same year, Huang et al. [153] used EAM interactions to establish the potential barrier to movement from fcc to hcp sites as 0.04 eV, in good agreement with the experiments. This work was continued in 2004 [122] and for jumps in the opposite direction, from hcp to fcc, a barrier of 0.03 eV was derived. For direct movement from fcc to fcc sites without passing over an intermediate hcp site, the barrier was huge, 0.37 eV. Based on those findings they concluded that the movement over the surface proceeded by a sequence of fcc-hcp-fcc jumps. Marinica et al. [154] made use of the EAM potential proposed by Mishin et al. [152] to find an energy barrier of 0.041 eV for copper atoms diffusing from fcc to hcp sites. The change in the potential energy during atom migration was previously shown in Fig. 3.26. Two things are of interest. The diffusion barrier is in good agreement with the experiments of Repp et al. [143] and Knorr et al. [144]. Furthermore, the fcc sites are slightly more favorable, by 0.005 eV, than the hcp sites. Also with EAM potentials, Ferrón et al. [155] did molecular dynamics simulations on a 14 layer slab with 270 atoms in each layer at different temperatures, from 100 K to roughly 800 K. For coordinated jumps, including long-distance transitions and rebounds, they found a good Arrhenius plot, shown in Fig. 5.33, giving a barrier of 0.036 ± 0.002 eV and Do = 1.73 × 10−4 cm2/sec. From

Temperature T (K) 1000 500 300 200 150

100

100

Dγ -all jumps Correlated jumps Uncorrelated jumps

1

1

Dγ (Å2/psec)

Diffusivity D(Å2/psec)

Temperature T (K) 1000 500 300 200 150

0.1

0.1

0

Fig. 5.33

50 1/kT (eV–1)

100

0.01 0

50 1/kT (eV–1)

100

Self-diffusion of Cu atoms on Cu(111) evaluated with EAM potentials. (a) Solid squares: from displacement and time. Open circles: from frequency of hopping. Solid line gives barrier of 0.030 ± 0.001 eV and Do = 1.56 × 10−4 cm2/sec. Inset shows surface cells and atom trajectory. (b) Solid circles: correlated jumps. Open squares: uncorrelated jumps. Solid line is Arrhenius plot for the former, yielding a barrier of 0.036 ± 0.002 eV and Do = 1.73 × 10−4 cm2/sec (after Ferrón et al. [155]).

300

Diffusion on two-dimensional surfaces

observations of the mean-square displacement as a function of the temperature they derived an activation energy of 0.030 ± 0.001 eV and Do = 1.56 × 10−4 cm2/sec. Bulou and Massobrio [68] calculated a self-diffusion barrier of 0.043 eV using a second-moment approximation to tight-binding. They also evaluated a barrier of 1.455 eV for self-diffusion by atom exchange. A kinetic Monte Carlo method and effective medium energetics were used by Jahma et al. [124] to evaluate a self-diffusion barrier of 0.026 eV, which of course is much too small. Durukanoglu et al. [117] used the nudged elastic band method and relied on EAM potentials to evaluate the diffusion barriers of single atoms on a flat Cu(111) surface using the local-density approximation. They calculated much too small a value of 0.01 eV for hopping; for exchange on this plane they found a much higher value of 1.42 eV. This work has been challenged by Kong and Lewis [69,126], who relied on the same EAM potentials, but calculated the diffusion parameters by summing over all atoms of the crystal. At an fcc site of the (111) plane they arrived at a diffusion barrier of 0.030 eV, and a prefactor of 1.84 × 10−4 cm2/sec; at an hcp site the barrier was 0.027 eV, and the prefactor 2.06 × 10−4 cm2/sec. They concluded that the local-density approximation was not generally valid for diffusion calculations. Kim et al. [70] have recently given barriers to atom jumps of 0.043 eV, and 1.352 eV for exchange, derived in molecular dynamics simulations with RGL interactions, in very reasonable agreement with experiments. Yang et al. [156] studied diffusion of clusters on a Cu(111) surface with MD and EAM potentials and for single atom movement derived a value of 0.06 eV and a prefactor of 3.01 × 10−4cm2/sec. Since 1996 the theoretical estimates are in general agreement with the experimental results, and it appears that for self-diffusion on Cu(111) the data are in excellent shape. It also seems that self-diffusion on Cu(111) is quite insensitive to long-range interactions – even calculations with Morse and Lennard-Jones potentials give reasonable results. Atom motion clearly proceeds by hopping; exchange involves huge barriers, and seems unlikely on close-packed surfaces. Only on stressed surfaces is there a chance for exchange. What is not quite clear, however, is how the two kinds of adsorption sites, fcc and hcp, influence the diffusion of adatoms.

5.8.2

Hetero-diffusion It should also be noted that there has been some work on hetero-diffusion. Van Siclen [130] performed calculations of the activation energy for hopping of indium on Cu(111) using EAM potentials and found a low value of 0.02 eV for the barrier. Exchange was unlikely, requiring 1.26 eV. The difference between binding at fcc and hcp sites was only 0.004 eV. In 1996 Breeman et al. [157] used embedded atom calculations to do Monte Carlo simulations for the same system and arrived at a higher activation energy of 0.08 eV. RGL potentials were used by Evangelakis et al. [138] to calculate diffusivities of gold atoms on Cu(111), with atoms hopping between fcc and hcp sites. From the Arrhenius plot shown in Fig. 5.34 they arrived at a barrier of 0.08 ± 0.01 eV and a prefactor of 3 × 10−4 cm2/sec for temperatures below 700 K. They also calculated the barrier for atom hopping by energy minimization and found a slightly smaller value of 0.061 eV, which they ascribed to including all types of jumps in the migration energy.

301

5.9 Molybdenum: Mo(110), (111), (100)

Temperature T (K)

1000 800

Diffusivity D(cm2/sec)

10–4

600

400

300

ED = 0.08±0.01 eV Do = 3 × 10–4 cm2/sec

10–5 10

15

20

25

30

35

40

1/kT (eV–1)

Fig. 5.34

Arrhenius plot for the diffusivity of gold on Cu(111), obtained using RGL potentials. Prefactor is 3 × 10−4 cm2/sec (after Evangelakis et al. [138]).

In 1997 Padilla-Campos and Toro-Labbé [158] looked at the diffusion of the alkali metals Li, Na, and K on Cu(111) using density-functional methods and the local-density approximation. Their data are highly doubtful, as they had only three layers in their simulation. They were able to distinguish between fcc and hcp sites. However their findings are a bit puzzling. Movement of Li from fcc to hcp involved a barrier of 0.20 eV with a prefactor of 6.49 × 10−5cm2/sec, from an hcp to a fcc site 0.054 eV and a prefactor of 4.54 × 10−5cm2/sec. For Na, movement between fcc to hcp required overcoming a barrier of 0.30 eV with a prefactor 5.32 × 10−5cm2/sec, and for an hcp to an fcc transition the barrier was 0.13 eVand the prefactor 5.21 × 10−5cm2/sec. For potassium the energy barrier from fcc to hcp was 0.20 eV and the prefactor 2.02 × 10−5cm2/sec; the opposite way cost 0.05 eV, with a prefactor 1.83 × 10−5cm2/sec. Kim et al. [70] relied on molecular dynamics simulations and RGL interactions to estimate the following barriers for atom hops on Cu(111): Ni, 0.045 eV, Pd 0.041 eV, Ag 0.034 eV, Pt 0.009 eV, and Au 0.039 eV. The last barrier is much lower than the value obtained by Evangelakis et al. [138]. The conditions of lead and copper alloying on the Cu(111) surface were studied by Anderson et al. [159] using the STM. To understand their measurement they calculated the diffusion barriers with VASP using the NEB method with the GGA approximation. For movement of lead atoms by hopping they estimated a barrier of ~0.03 eV while for exchange they calculated a much higher value of 1.17 eV. They concluded that intermixing in this system proceeded by vacancy mediated exchange. Hetero-diffusion on this plane definitely needs much more attention, both in experiments as well as theory, as there are no reports of experimental studies.

5.9

Molybdenum: Mo(110), (111), (100) Surprisingly there is only little information available, either experimental or theoretical, about self-diffusion on the densely packed (110) surface of molybdenum. This despite

302

Diffusion on two-dimensional surfaces

–12 ED = 0.54 eV Do = 8.7 × 10–4cm2/sec ln D

–13

–14

–15 8

Fig. 5.35

9

10 11 12 Temperature T(104 K)

13

Diffusivity of Cu on Mo film as a function of reciprocal temperature, as measured by secondary ion emission; based on the work of Abramenkov et al. [161].

the fact that molybdenum is a neighbor of tungsten, for which much has been done. However, there are some data that have been obtained for hetero-diffusion. Studies were done by Dhanak and Bassett [160] who examined rhodium diffusion with field ion microscopy and found an activation energy of 0.62 ± 0.06 eV, with a prefactor of 1 × 10−3 cm2/sec. This is a direct measurement of the movement of one atom deposited on the plane, which makes it quite reliable. Abramenkov et al. [161] looked at the spreading of copper atoms on a molybdenum ribbon of unspecified surface orientation, and under quite poor vacuum conditions (~10−6 Torr). From measurements at four temperatures they obtained an Arrhenius plot, shown in Fig. 5.35, which yielded an activation energy for diffusion of 0.54 eV and a prefactor Do = 8.7 × 10−4 cm2/sec, but it is clear that this result cannot be compared with single atom data. Saadat [162] a few years later did a field ion microscopic study of the motion of gallium, indium, and tin on Mo(110). There are only a few notes about the experimental conditions in this paper, but vacuum conditions seem to have been excellent. The field ion images look very good, but are a bit confusing, as the voltage on the tip is claimed to be very low (3.2 kV), but the photograph is of quite a large surface with an appreciable radius. Measurements of the movement of single atoms deposited at the surface at five temperatures gave an Arrhenius plot, shown in Fig. 5.36, with an activation energy of 0.25 eV for gallium with a prefactor Do = 2 × 10−5 cm2/sec, 0.30 eV for indium with Do = 3 × 10−5cm2/sec, and 0.50 eV for diffusion of tin atoms with Do = 4 × 10−5 cm2/sec. Jubert et al. [163] studied the nucleation of iron on Mo(110), using the scanning tunneling microscope to measure the island density formed at different temperatures at a coverage of 0.12 ML. They arrived at a diffusion barrier of 0.1 ± 0.05 eV and a frequency factor of ν0 = 8 × 1011 ± 1 sec−1. By comparison with films formed in molecular beam epitaxy they concluded that in pulsed laser deposition the diffusivity was four orders of magnitude higher, probably due to the higher energy of the impinging species. Movement of barium atoms on Mo(110) was investigated by Vedula et al. [164,165] who measured changes in the work function during spreading. For the lowest coverage

303

5.9 Molybdenum: Mo(110), (111), (100)

0 –5

In[/t]

–10 –15 –20 –25 Ga on Mo

–30

In on Mo

–35

Sn on Mo –40 3

Fig. 5.36

4 5 1000/T (K–1)

6

Arrhenius plot for the diffusivity of Ga, In, and Sn on Mo(110) plane, as measured by field ion microscopy. Diffusion barriers for Ga 0.25eV, In 0.30 eV, and Sn 0.50 eV (after Sadaat [162]).

Iog D (cm2/sec)

0.8 0.6 –7

0.4 0.2

0.0

Fig. 5.37

0.2

0.4 0.6 Coverage Θ

0.8

Activation Energy ED (eV)

D ED

–6

0.0 1.0

Diffusion characteristics for barium on Mo(110), determined from work function changes. Activation energy and diffusivity are given as a function of the coverage (after Naumovets et al. [165]).

of 1013 atoms/cm2 the value of the activation energy for diffusion amounted to 0.7 eV as indicated in Fig. 5.37 and the prefactor was 20 cm2/sec. These values probably do not reflect single atom movement. Not much else is known about diffusion on this plane, and more experimental work is clearly called for. The only two calculations for molybdenum surfaces were done quite early in the game, using Morse potentials without many-body interactions [47]. Flahive and Graham calculated a barrier of 0.4 eV for self-diffusion on Mo(110), a huge value of 1.9 eV for Mo(111) and for Mo(100) 1.4 eV, as shown in Fig. 4.9. Davydov, using the cohesion energy [49], estimated a diffusion barrier of 1.20 eV on Mo(110), a rather high value.

304

Diffusion on two-dimensional surfaces

5.10

Ruthenium: Ru(0001) In 1996, Hamilton [166], in his study of the dislocation mechanism of cluster diffusion, briefly mentioned an EAM value of 0.057 eV for movement of a single silver adatom on the ruthenium (0001) surface. The activation energy was in agreement with the bridge site energy and the prefactor was in the standard range; the mechanism of movement was not investigated.

5.11

Rhodium: Rh(111), (100)

5.11.1

Self-diffusion For growth on the face-centered cubic (111) plane, atoms must be added to the fcc sites, but also available are very similar hcp binding sites. However, on rhodium, no attempts have been made to determine which of these is filled by adatoms. The (111) surface is very smooth; rapid diffusion can be expected, and that was shown to be the case by the work of Ayrault [167] in Fig. 4.19, who from studies in 1974 by FIM at seven temperatures determined an activation energy of 0.16 ± 0.02 eV and a prefactor of 2 × 10−4 cm2/ sec on (111). The study was done with one adatom deposited on the surface and looked directly at the displacements, making the work quite reliable. In 1996, Tsui et al. [168] looked at the density of rhodium islands formed on Rh(111) at different temperatures and at a coverage of 0.1 ML. They used the STM and nucleation theory to analyze the data, assuming a critical size equal to one. From these measurements, shown in Fig. 5.38, they arrived at an activation energy of 0.18 ± 0.06 eV for diffusion. This is in quite good agreement with the work of Ayrault [167], and suggests that in this instance at least classical nucleation theory is valid.

10–2

N (Å–2)

10–3

10–4

ED = 0.18±0.06 eV

10–5 1.0

Fig. 5.38

1.5

2.0 2.5 1000/T (K–1)

Diffusion of Rh on Rh(111). Saturation density of islands as a function of reciprocal temperature, leading to a barrier of 0.18 ± 0.06 eV (after Tsui et al. [168]).

5.11 Rhodium: Rh(111), (100)

305

The structure of the (100) surface is quite different from (111), with deep recesses which can hold an adatom. On this surface an activation energy of 0.87 ± 0.07 eV, and a prefactor of 1 × 10−3 cm2/sec was reported by Ayrault [167] and the Arrhenius plot is shown in Fig. 4.19. These are quite early direct FIM studies, with only one adatom on the surface, but the results still seem to be valid. Somewhat more work has been done on the Rh(100) plane. Self-diffusion of Rh has been looked at briefly by Kellogg [169], also by FIM. From the mean-square displacement at 300 K, he derived a barrier of 0.83 ± 0.05 eV by assuming a standard value for the prefactor; this is reasonably close to the more detailed measurements by Ayrault [167]. Atoms moved by hopping on this surface; no exchange was observed. In 1994 Kellogg again looked at single atom motion and reported a barrier of 0.84 eV, assuming a standard prefactor. His measurements were done at one temperature, 296 K, and consisted of 70 observations [170]. The agreement in these studies is remarkable. Theoretical estimates of diffusion on rhodium surfaces began quite early. Using Morse potentials, Flahive and Graham [47] calculated a barrier of 0.05 eV for Rh(111) and 0.7 eV for Rh(100), as shown in Fig. 4.9. In 1983, McDowell and Doll [171,172] used a Lennard-Jones potential to do molecular dynamics simulations for diffusion. On Rh(100) they found a barrier of 0.87 ± 0.16 eV with a prefactor Do = 4.06 × (e±0.85 ) × 10−3 cm2/ sec. For Rh(111) they reported an activation energy of 0.27 ± 0.026 eV and a prefactor of 7.10 × (e±0.25 ) × 10−4 cm2/sec. Later, Voter and Doll [173] compared various evaluations of the activation energy for diffusion on Rh(111) and Rh(100) using Lennard-Jones potentials. With Monte Carlo simulation and transition-state theory (MCTST) they found a barrier of 0.224 ± 0.003 eVon Rh(111), with a prefactor of 5.4(×1.37±1) × 10−4 cm2/sec for a fit to low temperatures only. When they analyzed a wider range of temperatures they obtained a barrier of 0.20 ± 0.01 eV with a prefactor 2.91(×1.16±1) × 10−4 cm2/sec for the (111) plane [174]. In molecular dynamics simulations the results were 0.29 ± 0.03 eV for the diffusion barrier on (111) and 7.1(×1.28±1) × 10−4 cm2/sec for the prefactor. On Rh(100), these quantities were 1.03 ± 0.01 eV and 6.1(1.5±1) × 10−3 cm2/sec for low temperatures [171]; for a wider temperature range they got 0.92 ± 0.12 eV for the barrier with a prefactor of 3.7(×1.78±1) × 10−3 cm2/sec. A comparison of the data with the experiments of Ayrault [167] for low temperatures is shown in Fig 5.39; estimates over a wider range of data are given in Fig. 5.40. Voter also did a simple TST calculation, but the data are very similar to the MC investigations. In 1986, Voter [175] again used Lennard-Jones potentials to model cluster self-diffusion on Rh(100). Resorting to transition-state theory he came up with an activation energy of 1.094 eV for single Rh atoms. In 1991 Lynden-Bell [176] looked at self-diffusion on Rh(100) relying on molecular dynamics simulations using Sutton–Chen potentials. For hopping she arrived at an activation energy of 0.388 ± 0.002 eV, far from the experimental values, and a prefactor 5.6 × 10−3(×2±1) cm2/sec. Sanders and DePristo [76,177], in one of the first large applications of a many-body potential, used corrected effective medium theory to arrive at a self-diffusion barrier of 0.80 eV with a jump prefactor of 4.4 × 1012 sec−1 and Do = 3 × 10−3 cm2/sec on Rh(100); on Rh(111) the barrier proved to be 0.035 eV with a prefactor of 6.2 × 10−4 cm2/sec. Using molecular dynamics they found 0.092 ± 0.033 eV for the barrier and a prefactor 4.6 ± 2.3 × 10−4 cm2/sec on Rh(111). They also gave estimates

306

Diffusion on two-dimensional surfaces

0

0 Rh on Rh(100)

–20 (a)

–30

–30 –40

(b)

–50 MCTST EXPERIMENT

–40 –50

Rh on Rh(111)

–20 In D

In D

–10

Fig. 5.39

–10

0

1

2 1000/T

MCTST EXPERIMENT

–60 –70 3

–80

4

0

5

10 15 1000/T

25

Arrhenius plot for the diffusivity of Rh on Rh surfaces, calculated with Lennard-Jones potentials and compared with the experiments of Ayrault [167] (after Voter and Doll [173]). –6

–6 Rh on Rh(100)

Rh on Rh(111) –8 In D

In D

–8 –10 (a) MCTST

–12

–10 (b) MCTST

–12

Dynamics

Dynamics –14

–14 0

Fig. 5.40

20

5 100000/T

10

0

5

10

15

100000/T

Diffusion of Rh on Rh planes in its dependence upon reciprocal temperature, as obtained from L-J calculations. (a) On Rh(100), diffusion barrier of 1.03 ± 0.01 eV was derived from MCTST and data for dynamics [172]. (b) On Rh(111), a barrier of 0.29 ± 0.03 eV was obtained (after Voter and Doll [174]).

with Morse and Lennard-Jones potentials on this plane by doing molecular dynamics. For the first of the two potentials they gave for the barrier 0.118 ± 0.043 eV and Do = (7.1 ± 4.5)×10−4 cm2/sec; with Lennard-Jones potentials they found a barrier of 0.234 eV with Do = 4.5 × 10−4cm2/sec, previously also calculated by Mruzik and Pound [178]. Using molecular dynamics simulations with the Sutton–Chen potential, Shiang [60,102] achieved a barrier of 0.832 eV on Rh(100) with a prefactor of 1.36 × 10−3 cm2/sec. With Morse potentials he found 0.799 eV and 2.13 × 10−3 cm2/sec. On Rh(111) these quantities proved to be 0.106 eV for the barrier and 6.09 × 10−4 cm2/sec for the prefactor. Using Morse potentials the barrier obtained was 0.089 eV and the prefactor 6.40 × 10−4 cm2/sec, as shown in Fig. 5.41. It appears there is no big difference between barriers calculated with many-body and pair potentials. Kumar et al. [108] used a smart Monte Carlo technique with Lennard-Jones potentials to arrive at a barrier of 0.27 ± 0.01 eV and a prefactor of 1.1 ± 0.2 × 10−5 cm2/sec for rhodium on Rh(111). The barrier was quite large compared with experimental findings, and no information about movement between fcc and hcp sites was derived. Feibelman

307

5.11 Rhodium: Rh(111), (100)

–7.8

–10 Rh on Rh(100)

–11

Morse potential SC potential

–13

–8.4 –8.6

–14

–8.8

–15

(a)

–16 100

Fig. 5.41

Morse potential SC potential

–8.2 In D

In D

–12

Rh on Rh(111)

–8.0

(b) 150

200 250 1/T

300

350

–9.0 100

150

200

250 300 1/T

350

400

Diffusion of Rh adatoms on Rh planes, with the temperature dependence calculated by the Sutton– Chen as well as with the Morse potential. (a) On Rh(100), barrier is 0.832 eV from SC and 0.799 eV from Morse. (b) On Rh(111) the barrier was lower, 0.106 eV from SC and 0.089 eV from Morse calculations (after Shiang [60,102]).

and Stumpf [179], working with density-functional methods on Rh(100), found a selfdiffusion barrier of 0.89 eV for atom hopping and > 1.11 eV for atom exchange with the substrate. This barrier is in excellent agreement with the experiments of Ayrault [167]. For Rh(111), Davydov [49] estimated a barrier of 0.26 eV, and 0.44 eV for the (100) plane, in both cases using his cohesion approximation. It is not unusual for this approach to describe the nature of adatom movements incorrectly. Ma´ca et al. [180,181] calculated diffusion energies in various environments and arrived at an energy barrier of 0.15 eV for self-diffusion on Rh(111) using RGL potentials. Agrawal et al. [26] with Lennard-Jones potentials found a diffusion barrier of 1.02 eV and a prefactor Do = 3.0 × 10−3 cm2/sec on Rh(100), and 0.21 eV and Do = 1.9 × 10−4 cm2/sec for Rh(111). Chang and Wei [28] in 2005 obtained a barrier of 0.99 eV for atom hopping on Rh(100) using ab initio densityfunctional methods. For atom exchange the barrier was considerably higher, 1.20 eV. The theoretical estimates for diffusion on Rh(111) have a significant spread, making a choice difficult, but for the (100) plane they are quite close together and in reasonable accord. All investigations seem to agree that the mechanism of movement on the Rh(100) surface is atom hopping, not exchange. On Rh(111), however, not much attention has been paid to the two kinds of adsorption sites, fcc and hcp. The experimental findings for this material seem to be in quite good shape, even if made relatively early, partly because direct measurements are available.

5.11.2

Hetero-diffusion Kellogg [182,183] in 1993 briefly looked at platinum on Rh(111) in the field ion microscope, and found Pt to be mobile at 77 K, the temperature at which he did imaging. From this he gave an upper limit of 0.22 eV for the barrier, assuming a prefactor Do = 1 × 10−3 cm2/sec. FIM measurements at an imaging temperature close to 20 K will be desirable to get more precise estimates. Kellogg [182] also looked more intensively at the diffusion of platinum atoms on Rh(100). Although based on only four temperatures,

308

Diffusion on two-dimensional surfaces

Table 5.2 Diffusion on Rh(100) [89]. Frequency Prefactor (×1012 sec−1)

Diffusion Barrier (eV)

Adatom Bridge Hops

Exchange

Hopping

Exchange

Ni Pd Cu Ag

6.50 8.64 3.62 4.14

0.75 0.73 0.53 0.51

1.21 1.46 1.44 1.91

5.82 4.07 2.46 3.31

–36 Pt on Rh(100)

In [/4t] (cm2/sec)

–37 –38 –39 –40 –41

ED = 0.92 eV Do = 2 × 10–3cm2/sec

–42 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 1/T (K–1) Fig. 5.42

Arrhenius plot for the diffusivity of Pt on Rh(100), obtained by field ion microscopy (after Kellogg [183]).

the activation energy of 0.92 ± 0.13 eV and prefactor of 2.0 × 10−3 ± 1.9 cm2/sec derived from the Arrhenius plot in Fig. 5.42 appear reasonable. The displacements of platinum during diffusion show a (1 × 1) map, suggesting that diffusion occurred by hopping. This is different from the behavior of iridium atoms, which diffuse on the Rh(100) surface by atom exchange [169]. In 1992, Sanders and DePristo [76] carried out transition-state theory calculations of hetero-diffusion on the Rh(100) plane, examining hopping of Ni, Cu, Pd, Ag, Pt, and Au with potentials from MD/MC-CEM. Activation energies were as follows: nickel 0.76 eV, copper 0.74 eV, palladium 0.66 eV, silver 0.50 eV, platinum 0.66 eV, and gold 0.55 eV. The frequency prefactors were all in the range 1012 sec−1. The multiples of this factor were as follows: nickel 5.5; copper 5.4; palladium 4.1; silver 4.9; platinum 3.7; gold 2.7. The value obtained for platinum movement is lower than that experimentally derived by Kellogg [182]. Two years later, Perkins and DePristo [89], relying on the same approach, compared hopping with atom exchange. The results are listed in Table 5.2. These estimates were made with 226 movable atoms in the first three layers, 81 in the first and third, and 64 in the second, but despite that hopping appeared as the favored mechanism for diffusion. Exchange was experimentally found [169] for diffusion of iridium atoms, but Perkins and DePristo unfortunately did not do calculations for this system. As is clear from this survey the hetero-diffusion effort has been much too small.

5.12 Palladium: Pd(100)

5.12

Palladium: Pd(100)

5.12.1

Self-diffusion

309

The first measurements of self-diffusion on Pd(100) were made by Flynn-Sanders et al. [184], who in 1993 did LEED studies of homo-epitaxial growth. From the temperature for the onset of diffusion they estimated a barrier ranging from 0.52 eV to 0.61 eV, assuming a frequency prefactor ν0 = 1013 sec−1. A second analysis of the LEED data yielded 0.59 eV to 0.63 eV, assuming a frequency prefactor of 5 × 1012 sec−1 [185]. From the ring diameter in the nucleation and growth region together with Monte Carlo simulations a barrier between 0.51 eV and 0.62 eV was deduced, assuming a critical size equal to one. The data were reexamined by Evans and Bartelt [186], who arrived at a barrier of  0.65 eV assuming ν0 = 5 × 1012 sec−1. The mechanism of movement was not derived in these studies. Using Morse potentials, Flahive and Graham [47] calculated a barrier of 0.55 eV, as shown in Fig. 4.9. Calculations with many-atom potentials were started by Liu et al. [4], who relied on the embedded atom method and arrived at a barrier for atom hopping of 0.71 eV and a prefactor of 1.2 × 10−3 cm2/sec in the AFW approximation; with VC potentials they reported a barrier of 0.74 eV and a prefactor Do = 6.0 × 10−3 cm2/sec. They also investigated the exchange mechanism on this plane and found slightly lower activation energies, 0.61 eV (AFW) and 0.59 eV with a prefactor of 3 × 10−2 cm2/sec (VC). Sanders and DePristo [76] in 1992 relied on MD/MC-CEM and transition-state theory to evaluate a barrier of 0.61 eV with a frequency prefactor of 4.2 × 1012 sec−1, but considered only atom hopping. Using CEM estimates, Perkins and DePristo [77] found a static energy barrier of 0.64 eV, and 0.75 eV by molecular dynamics. The barrier for diffusion by adatom exchange proved considerably higher, 1.03 eV with CEM and 1.38 eV using MD/MC-CEM. It must be noted here, however, that in the first substrate layer only 13 atoms were movable, 9 in the first layer and 4 in the second, and this may have contributed to the high barriers evaluated for atom exchange. In 1995 Perkins and DePristo [29] continued work on the same system with a larger number of movable atoms and found that on increasing the number of movables from 13 (appropriate for the previous work) to 226, the activation energy for the exchange mechanism decreased from 1.38 eV to 0.96 eV with MD/MC-CEM potentials and from 1.03 eV to 0.70 eV using CEM. The hopping mechanism was still slightly favored energetically, but the difference was now much smaller; the size of the used cell is crucial in attaining a correct description of nature, as is apparent in Fig. 5.43. Stoltze [61] calculated activation energies for hopping in self-diffusion from effective medium theory and derived a barrier of 0.503 eV. Boisvert et al. [78] also did embedded atom calculations and reported an activation energy of 0.51 eV, with a prefactor of 3.41 × 1013 sec−1 for atom exchange and 0.7 eV and a prefactor 5.51 × 1013 sec−1 for atomic jumps. This study suggests exchange as the leading mechanism. However, the surface layer of the substrate had only 72 atoms in it. Also with embedded atom potentials, Mehl et al. [82] found a barrier of 0.71 eV in self-diffusion by hopping; exchange was not considered. Feibelman and Stumpf [179] did density-functional

310

Diffusion on two-dimensional surfaces

0.4

sum of squared displacement (bohr2)

sum of squared displacement (bohr2)

7 Bridge x - direction y - direction z - direction

(a)

0.5

0.3 0.2 0.1 0.0

Fig. 5.43

100 200 300 active substrate atoms

0

400

(b) Atomic Exchange x - direction y - direction z - direction

6 5 4 3 2 1

0

100 200 300 active substrate atoms

400

Sum of squared displacements in self-diffusion as a function of the number of atoms active for Pd(100); calculations by corrected effective medium method. (a) For ordinary atom hopping. (b) For diffusion by atom exchange (after Perkins and DePristo [29]).

In[Γ (×1011 sec–1)]

1.00000 ED = 0.62±0.04 eV

0.36788 0.13534 0.04979 0.01832 0.00674 1.1

1.2

1.3

1.4

1.5

1.6

1.7

1000/T (K–1) Fig. 5.44

Arrhenius plot for the self-diffusion of Pd on Pd(100), calculated by embedded atom method (after Evteev et al. [187]).

calculations for self-diffusion on Pd(100). They found a barrier of 0.71 eV for atom jumps, and 0.82 eV for diffusion via atom exchange. Using his cohesion approximation, Davydov [49] estimated an activation energy of only 0.15 eV, a value which appears much too low. Agrawal et al. [26] worked with Lennard-Jones potentials and found a barrier for jumps of 0.60 eV with a prefactor of 1.1 × 10−3 cm2/sec. Evteev et al. [187] relied on EAM potentials to do molecular dynamics simulations. From the temperature dependence of the number of elementary events they obtained the Arrhenius plot of Fig. 5.44 and derived an activation energy of 0.62 ± 0.04 eV. Exchange was not considered. For hopping, Chang and Wei [28] calculated an energy barrier of 0.87 eV using the VASP [22–24] simulation package. For atom exchange, the barrier proved to be comparable, 0.85 eV, so that it is difficult to figure out which mechanism is favorable, or whether both are present at the surface. Recently, Kim et al. [70] carried out molecular dynamics simulations using RGL potentials to find the barrier to jumps amounted to 0.621 eV, whereas exchange required 0.725 eV.

311

5.12 Palladium: Pd(100)

The results for diffusion on palladium, both from experiments and from calculations, are not in any sort of agreement. Self-diffusion barriers range from 0.5 eV to 0.75 eV for hopping and from 0.6 eV to 0.9 eV for atom exchange. At the moment it is difficult to say which mechanism is preferred in diffusion or if they coexist at the surface. Better experiments are clearly in order here.

5.12.2

Hetero-diffusion In 1989, Evans et al. [188] looked at the growth of a Pt film on Pd(100) with LEED. From the temperature for the onset of diffusion, around 150 K, they deduced an activation energy around 0.43 eV, assuming a frequency prefactor of 1013 sec−1. There were no attempts made to detect the mechanism of movement. Félix et al. [189] carried out helium scattering experiments on silver deposits on Pd(100). The intensity specularly reflected was measured to determine the particle size distribution, assumed to arise from the random motion of the silver atoms, as a function of time and temperature. From the Arrhenius plot of the jump rate, shown in Fig. 5.45, they derived an activation energy for silver diffusion of 0.37 ± 0.03 eV, with a somewhat small frequency prefactor of 8 × 109 sec−1. They also calculated a static barrier with embedded atom theory and found a much higher value of 0.66 eV. A year earlier, the same authors [190] reported for the same system a barrier of 0.35 ± 0.03 eV and a frequency prefactor 2.7 × 109 sec−1. Massobrio et al. [191] did molecular dynamics simulations using EAM potentials to explain the He scattering results for the Ag/ Pd(100) system. For atomic movement they found a hopping barrier in the direction of 0.6 eV and an exchange barrier in the direction of 0.75 eV. Exchange was clearly not the favored mechanism. Sanders and DePristo [76] in 1992 relied on MD/MC-CEM and transition-state theory to evaluate barriers for hetero-diffusion on Pd(100), with the following results: activation energy for nickel 0.71 eV, frequency prefactor 5.2 × 1012 sec−1; copper 0.72 eV, 5.3 × 1012 sec−1; rhodium 0.75 eV, 4.3 × 1012 sec−1; silver 0.42 eV, 8.0 × 1012 sec−1; platinum

In[Γ (sec–1)]

4

Ag/Pd(100)

2 0 –2 ED = 0.37±0.03 eV ν0 = 8 × 109 sec–1

–4 4.4

Fig. 5.45

4.6

4.8

5.0 5.2 5.4 1000/T (K–1)

5.6

5.8

6.0

Diffusivity of Ag adatom on Pd(100) in its dependence upon reciprocal temperature, determined by thermal atom scattering (after Félix et al. [189]).

312

Diffusion on two-dimensional surfaces

Table 5.3 Diffusion on Pd(100) [89]. Frequency Prefactor (×1012 sec−1) Diffusion Barrier (eV) Adatom Bridge Hops

Exchange

Hopping

Exchange

Rh Ni Cu Ag

13.76 9.74 3.98 6.38

0.75 0.70 0.52 0.50

0.70 0.78 0.98 1.34

7.85 7.01 2.63 4.96

0.64 eV, 4.0 × 1012 sec−1; gold 0.54 eV, 2.8 × 1012 sec−1. Perkins and DePristo [89] reconsidered diffusion on Pd(100) and now arrived at the activation energies listed in Table 5.3. In this table the frequency prefactors for the exchange process were roughly twice as high as for hopping, but still in the range 1012 sec−1. Only for rhodium adatoms were conditions favorable for diffusion by atom exchange. The barrier for silver on Pd(100) appears to be a bit higher than the experimental findings by Félix et al. [189] but comparable with the results of Massobrio et al. [191]. It must be noted here, however, that in the first substrate layer of Perkins and DePristo [89] 226 atoms were movable. Estimates for hetero-atom diffusion were made by Kim et al. [70] using molecular dynamics based on RGL potentials. The values were Ni 0.566 eV, Cu 0.558 eV, Ag 0.459 eV, Pt 0.986, and Au 0.590 eV. These barriers are not in very good agreement with the work of Perkins and DePristo [89].

5.13

Palladium: Pd(111)

5.13.1

Self-diffusion Experimental studies of self-diffusion on Pd(111) were made in 2000. Steltenpohl and Memmel [192,193], used the STM to observe nucleation and growth for 0.08 of a monolayer of palladium deposited at 210 K and at 420 K. Their results for the island density at various temperatures, shown in Fig. 5.46, were interpreted in accord with classical nucleation theory. This led to a diffusion barrier of 0.35 ± 0.04 eV. More surprising is the jump prefactor, ν0 = 6 × 10 16 ± 0.6 sec−1, four orders of magnitude above the normally expected value. The authors relate these results to the electronic structure of Pd(111), but these findings still need to be checked. Unfortunately, this study stands alone and no other direct measurements of adatom movement are available for palladium surfaces. Theoretical investigations with Morse potentials were carried out by Flahive and Graham [47], who estimated a barrier of 0.05 eV, as shown in Fig. 4.9. Calculations with many-atom potentials started in 1991 by Liu et al. [4], who relied on the embedded atom method with AFW and VC potentials. The results obtained were 0.031 eV for the barrier and 5 × 10−4 cm2/sec for the prefactor (AFW), and 0.059 eV together with 4.5 × 10−4 cm2/sec (VC).

313

5.13 Palladium: Pd(111)

(a)

(b)

500 Å

500 Å Temperature T(K) 420

nx = number of island per site

10–4

340 300

260

220

(c) Pd/Pd(111)

10–5

10–6 25

Fig. 5.46

ED = 0.35 ± 0.04 eV ν0 = 6 × 1016 ± 0.6 sec–1

30

35

40 45 50 1/kT (eV–1)

55

60

Self-diffusion of Pd atoms on Pd(111) examined by STM. Images of 0.08 ML deposited at (a) 210 K and (b) 420 K. (c) Dependence of diffusivity upon reciprocal temperature (after Steltenpohl and Memmel [192,193]).

Stoltze [61] used effective medium theory to estimate a barrier of 0.104 eV. Mortensen et al. [63] used density-functional theory to arrive at a barrier of 0.21 eV with LDA and 0.17 eV relying on GGA. They also found that binding at an fcc site was 0.13 eV stronger than at an hcp. In 1996, Li and DePristo [62] with corrected effective medium theory found for hopping on this plane an activation energy of 0.034 eV. Davydov [49] estimated an activation energy of 0.10 with his cohesive approach. Chang et al. [35] did calculations using embedded atom interactions and came up with a barrier of 0.04 eV, a value similar to the previous findings. Lennard-Jones potentials were used by Agrawal et al. [26] to estimate a diffusion energy of 0.14 eVand a prefactor 1.6 × 10−4 cm2/sec. At much the same time Papanicolaou [194] carried out molecular dynamics simulations at temperatures from 150 K to 900 K for self-diffusion on Pd(111), relying on RGL potentials. The results are given in the Arrhenius plot of Fig. 5.47. Here the data at temperatures less than 500 K are of primary

Diffusion on two-dimensional surfaces

Diffusivity D(10–6 cm2/sec)

314

ED = 0.127 eV

100.0

10.0

1.0 ED = 0.097 eV 0.1

0

20

40

60

80

1/kT (eV–1) Fig. 5.47

Dependence of diffusivity of Pd on Pd(111) upon reciprocal temperature, calculated using RGL potentials. Data at 500 K refer to nearest-neighbor jumps (after Papanicolau et al. [194]).

interest, and refer to nearest-neighbor hops. These data lead to a diffusion barrier of 0.097 eV and a prefactor Do = 4.4 × 10−4 cm2/sec. This barrier is much lower than the value from experiments. At higher temperatures, 500–900 K, he found a barrier of 0.127 eV and a prefactor 9.1 × 10−4 cm2/sec, which he attributed to long and correlated jumps. Recently, Kim et al. [70] carried out molecular dynamics simulations using RGL potentials to find that jumps of Pd atoms occurred over a barrier of 0.109 eV, exchange required an energy of 1.315 eV. As in all their other estimates, Kim et al. did not distinguish between fcc and hcp sites on the fcc(111) plane. The state of the Pd(111) plane is not good. Theoretical estimates for the activation energy lie in the range from 0.03 eV to 0.21 eV, and it is difficult to conclude anything. No information about how the existence of two adsorption sites, fcc and hcp, influence movement is given in the the literature. It is clear that further work is needed to provide a reasonable perspective about diffusion on palladium.

5.13.2

Hetero-diffusion Hunger and Haas [195] have examined the diffusion of 111In on Pd(111) at different temperatures using perturbed angular correlation spectroscopy. Indium atoms were already mobile at 100 K. Assuming a frequency prefactor of 1012 sec−1, they arrived at an activation energy of 0.23 ± 0.03 eV for the diffusion of 111In at a concentration of 10−4 monolayers. Due to the very low coverage it is likely that this study detected single atom movement, but unfortunately the technique is not capable of defect control. Diffusion of potassium on Pd(111) at a coverage of < 0.04 monolayer was studied using photoemission electron microscopy by Ondrejcek et al. [196]. They arrived at a value of < 0.130 eV, assuming a frequency factor of 1.7 × 1014 sec−1. The diffusion of potassium in the limit of zero coverage has been examined on Pd(111) by Sna´bl et al. [197] in 1996, who used photoelectron emission microscopy to measure the

315

5.14 Silver: Ag(111)

–12 ED = 0.066 ± 0.011 eV

–13 In [D(cm2/sec)]

Do = 1.3 ± 10–4.0±0.4cm2/sec –14 –15 –16

–17 5

6

7 1000/T

Fig. 5.48

8

9

10

(K–1)

Surface diffusivity of potassium atoms on Pd(111) as a function of reciprocal temperature, at a coverage < 0.05 ML. Triangles: from concentration profiles. Closed circles: from contour plots (after Sna´bl et al. [197]).

potassium concentration. Their results for a coverage lower than 0.05 ML, in Fig. 5.48, lead to an activation energy for diffusion of 0.066 ± 0.011 eV, and a prefactor Do = 1.3 × 10−4 ± 0.4 cm2/sec. The authors ascribe what they consider as a large uncertainty in the diffusion barrier to the presence of defects on the surface not detected by their instrument. Estimates for jumps in hetero-atom diffusion were made by Kim et al. [70] with molecular dynamics based on RGL potentials. The barrier for Ni was 0.087 eV, for Cu 0.091 eV, for Ag 0.074 eV, for Pt 0.127 eV, and for Au 0.110 eV. It appears, however, that not enough experimental work has been done to come to any firm conclusions about hetero-diffusion on palladium.

5.14

Silver: Ag(111)

5.14.1

Self-diffusion Among the first attempts to study self-diffusion on silver was the work of Stark [198], who measured the resistivity changes of thin, (111) oriented films of silver on which silver atoms were deposited. From measurements over a range of temperatures he found an activation energy of 0.58 eV for self-diffusion, a value much higher than what has been reported later and unlikely to represent single atom movement. Jones et al. [199] studied the deposition of silver on a W(110) surface in 1990. In their experiments, two layers of silver were formed first in the (111) orientation; only then did nucleation and growth of silver islands occur. This was examined with UHV scanning electron microscopy and interpreted by nucleation theory using kinetic rate equations to yield an activation energy for surface diffusion of 0.15 ± 0.10 eV. In calculations with effective medium theory they found a barrier of 0.12 eV. Venables [200] also analyzed measurements by Spiller et al. [201] on nucleation on top of two silver layers deposited

316

Diffusion on two-dimensional surfaces

on a W(110) surface and derived a barrier of ~0.1 eV. No possible strain factor was taken into account in the above studies. A few years later, in 1995, Luo et al. [202] carried out studies of silver on the (111) plane of silver, using SPA-LEED to measure the density of islands at different surface temperatures. From the average terrace size for 0.5 monolayers of silver and nucleation theory, with the critical nucleus size equal to one, they deduced a diffusion barrier of roughly 0.18 eV from their data, shown in Fig. 5.49; the coverage may, however, have been too big for single atom diffusion. Two years earlier, Henzler et al. [141] reported SPA-LEED measurements of the temperature dependence of the island density on Ag(111) at quite a high coverage of 0.5 ML, which yielded a barrier of 0.12 eV. Rosenfeld et al. [203] also examined the epitaxy of silver on Ag(111), using thermal energy He atom scattering from 0.05 monolayers of silver. Their results, in Fig. 5.50, Deposition temperature T(K)

Average terrace length (in atomic distances)

250 225

175

150

125

40

Θ = 0.5 ML ED = 0.18 eV 20

4

Fig. 5.49

200

5

6 1000/T (K–1)

7

8

Dependence of average terrace size of silver on Ag(111) upon reciprocal temperature (after Luo et al. [202]). Temperature T(K) 250 200

10–2

150

125

100

[1–√l/l0]2

Ag/Ag(111) 0.05 ML

10–3

ED = 0.051 ± 0.024 eV 10–4

3

4

5

6

7

8

9

10

11

1000/T (K–1) Fig. 5.50

Arrhenius plot of quantity proportional to density of Ag islands obtained from He intensity scattered from Ag(111) (after Rosenfeld et al. [203]).

5.14 Silver: Ag(111)

317

interpreted according to nucleation theory, led to a diffusion barrier of 0.051 ± 0.024 eV. At roughly the same time, Brune et al. [204] looked at this system with a STM. Relying on the nucleation theory expression for the island density, they arrived at a value of 0.097 ± 0.010 eV for the barrier to diffusion on Ag(111) and a frequency prefactor of ν0 = 2 × 1011 ± 0.5 sec−1. For silver deposition on one monolayer of silver on Pt(111), Brune et al. [204] made the same sort of measurements, and found a diffusion barrier of 0.060 ± 0.010 eV and a frequency prefactor of ν0 = 1 × 109 ± 0.6 sec−1 as shown in Fig. 3.69. They attributed the lower value of the activation energy to the effect of strain arising when silver was deposited on Pt(111), but did not comment about the frequency factor, which was at least two orders of magnitude too low. Brune et al. [204], in explaining findings from saturation island density measurements, ran calculations with EMT interactions and arrived at a lower value of 0.067 eV for silver atoms on silver (111) (compared with the 0.097 eV they derived from experiments). For silver on one monolayer of silver on Pt(111) their experiments indicated a barrier of 0.060 ± 0.010 eV, whereas the calculations yielded a value of 0.050 eV. In 2000, Schlösser et al. [142] examined island diffusion with a variable temperature STM, and to explain the experimental finding employed the ARTwork computer program of Stoltze [61] using EMT potentials. For movement of a single Ag atom on the Ag(111) surface their calculations led to a barrier of 0.068 eV. Cox et al. [205] looked at the shape of silver islands on Ag(111) with the scanning tunneling microscope and modeled the data by Monte Carlo simulations relying on EAM interactions. In their model, the atom sits in an fcc site and moves between fcc sites, with an hcp site intervening. On a silver layer of 0.3 monolayers the activation energy for self-diffusion by hopping was 0.1 eV with a prefactor of 1011 sec−1. This value is comparable with the findings of Brune et al. [204]. There is a substantial body of theoretical work dealing with the diffusion of silver atoms on silver surfaces started with the work of Flahive and Graham [47]. Using Morse potentials, they calculated a barrier of ~0.01 eV, shown in Fig. 4.9. Rilling et al. [206] in 1990 used EAM potentials to evaluate the energy changes occurring during atom jumps; they found a barrier height of 0.058 eV with a frequency prefactor of 1.4 × 1012 sec−1. Further work with EAM potentials was done by Liu et al. [4]. The barrier for AFW calculations was 0.059 eV and a prefactor of 5 × 10−4 cm2/sec, whereas using VC values the barrier was smaller, 0.044 eV, with a prefactor 4.1 × 10−4 cm2/sec. Sanders and DePristo [177] made estimates in 1992 using the MD/MC-CEM approximation and came up with a barrier of 0.023 eV and a prefactor of 6.2 × 10−4 cm2/sec in transition-state theory; with molecular dynamics they found the activation energy for diffusion to be 0.055 ± 0.011 eVand a prefactor of (6.9 ± 2.2)×10−4 cm2/sec. Using Morse potentials the results for the barrier were much smaller, 0.005 eV for transition-state theory, with a prefactor of 5.5 × 10−4 cm2/sec, and 0.039 ± 0.009eV with a prefactor of 4.7 ± 1.4 × 10−4 cm2/sec from molecular dynamics. Nelson et al. [207] carried out calculations using EAM, but with FBD set Universal 3 [57]. The barrier proved to be 0.06 eV. The calculations were done on a static lattice and

318

Diffusion on two-dimensional surfaces

considered only hopping. Stoltze [61] relied on EMT, gave an activation energy for selfdiffusion of 0.064 eV. Full-potential linear-muffin-tin-orbital estimates were made by Boisvert et al. [208] and found an energy barrier of 0.14 ± 0.02 eV, higher than previously reported values. They claimed the difference between fcc and hcp sites was very small and insignificant, justifying why they assumed that both sites were equal. In the same year, Boisvert et al. [78] used EAM and molecular dynamics simulations to validate the Meyer–Neidel rule for jumps on the Ag(111) surface; they reported a much lower barrier of 0.055 eV and a prefactor of 2.29 × 1012 sec−1, as shown in Fig. 5.51. Molecular dynamics simulations were made in 1995 using RGL potentials by Ferrando and Tréglia [209], who arrived at an activation energy of 0.068 eV and a prefactor of 2.6 × 10−4 cm2/sec from the Arrhenius plot in Fig. 5.52; the difference in the potential energy at fcc and hcp sites was 0.009 eV.

D (Å2/psec) or Γ (THz)

101 Γ (MD) D (MD) Γ (TST) 100

10–1 Ag(111) 10–2

Fig. 5.51

0

40 1/k T (eV–1)

80

Arrhenius plot for diffusivity of Ag adatom on Ag(111), obtained by molecular dynamics simulations using EAM potentials. Activation energy was 0.055 eV (after Boisvert et al. [359]). 0 Ag(111)

–1

log D

–2 –3 –4 Au(111) –5 –6 10

20

30 1/T

Fig. 5.52

40

50

60

(eV–1)

Dependence of the diffusivity of Ag on Ag(111) and Au on Au(111), calculated with RGL interactions, upon reciprocal temperature. Diffusion barrier of 0.12 eV for Au and 0.068 eV for Ag (after Ferrando and Tréglia [209]).

319

5.14 Silver: Ag(111)

Diffusivity D(cm2/sec)

Li and DePristo [62] also made estimates using MD/MC-CEM theory, to obtain a low barrier of 0.02 eV. Calculations with density-functional theory were reported by Mortensen et al. [63] in 1996. They found a barrier of 0.14 eV with the local-density approximation and 0.10 eV using the generalized gradient method, but did not see any difference for fcc and hcp sites. In 1997 Ratsch et al. [210] used density-functional calculations in the local-density approximation to arrive at a barrier of 0.081 eV for jumps. For movement of Ag atoms on one ML of Ag on Pt(111) they found a barrier of 0.063 eV, in good agreement with the calculations of Brune et al. [204], although they worked with a very small (2 × 2) cell. A year later, Ratsch and Scheffler [211], using functional-density theory, reported a slightly higher barrier of 0.082 eV and a frequency prefactor of 0.82 × 1012 sec−1. They also looked at silver moving on one monolayer of silver deposited on Pt(111) and found an activation energy of 0.06 eV and a frequency prefactor of 5.9 × 1012 sec−1. Papanicolaou et al. [212] used molecular dynamics simulations with a many-body potential from the second-moment approximation to the tight-binding model over a wide range of temperatures. They found that movement proceeds by adatom hopping on the surface, but there are two temperature regimes. Their results are presented in Fig. 5.53. At low temperatures of 200 K to 500 K, the activation energy for moving an adatom was 0.069 ± 0.001 eV; at higher temperatures of 500 K to 900 K the activation energy increased to 0.098 ± 0.006 eV. The static barrier for the lower temperature range was 0.063 eV. The increase in activation energy at higher temperatures was associated with longer and correlated jumps on the surface. However, the recent experimental study of Antczak [213] on W(110) showed an increasing number of long jumps with increasing temperature, but did not reveal an increase in the effective activation energy due to a correlation between the number of single and long jumps. In 1999, with his cohesion approximation, Davydov [49] estimated a self-diffusion barrier of 0.04 eV. A year later, Chang et al. [35] also worked out the same barrier of 0.04 eV using embedded atom interactions. In 2000 Baletto et al. [214] looked at the movement of silver atoms on a silver cluster, a Wulff polyhedron with (111) and (100) planes; they relied on molecular dynamics simulations using RGL potentials. One

Ag/Ag(111)

10–4 ED = (98±6) meV 10–5

ED = (69±1) meV –6

10

0

20

40

60

1/kT (eV–1) Fig. 5.53

Arrhenius plot for the diffusivity D of Ag atoms on Ag(111), derived by molecular dynamics simulations using RGL potentials (after Papanicolaou et al. [212]).

320

Diffusion on two-dimensional surfaces

polyhedron for which diffusion was studied consisted of 201 atoms, with a (100) plane of 9 atoms. The activation energy for movement from fcc to hcp sites on the (111) plane of the Ag polyhedron was found as 0.07 eV. Depending on Lennard-Jones potentials, Agrawal et al. [26] made estimates for a range of fcc materials. On Ag(111) the barrier was only 0.10 eV with a prefactor 1.4 × 10−4 cm2/sec. Chvoj et al. [215] relied on EAM interactions to do Monte Carlo simulations to arrive at an activation energy of 0.061 eV for atom hopping. Kong and Lewis [69] in 2006 calculated the diffusivity prefactor Do relying on EAM potentials and summing vibrations over all atoms in the system. Starting from an fcc site they obtained 2.78 × 10−4 cm2/sec, from an hcp site the prefactor was 2.80 × 10−4 cm2/sec, both independent of temperature from 300 to 600 K. Molecular dynamics simulations based on RGL interactions were carried out by Kim et al. [70] to obtain diffusion barriers on Ag(111). Atom jumps on this surface required an energy of 0.064 eV, whereas exchange occurred over a much higher barrier of 1.126 eV, results of the same sort as obtained in previous calculations. The one thing clear about self-diffusion on the (111) plane of silver is that there is no close agreement between the different experimental results that have been obtained for this system. That is disturbing, since all the results relied on nucleation theory. However, a value around 0.1 eV seems to be the most probable. Unfortunately no direct measurements of hopping rates were done on this plane to clarify the situation. The theoretical results also scatter widely, from 0.14 eV to 0.02 eV, and appear to be lower than what seems to emerge from the experiments, 0.18 eV to 0.05 eV, after excluding the results of Stark. However the range of theoretical and experimental findings is similar.

5.14.2

Hetero-diffusion The effect of antimony on the growth mode of silver on Ag(111) was examined in detail in 1994 by Vrijmoeth et al. [216], in a continuation of a study begun earlier. From the island density, which increased exponentially with the amount of antimony deposited, they concluded that antimony increases the activation energy of silver diffusion linearly. The increase in the diffusion barrier above that on the clean (111) surface is suggested in Fig. 5.54, but they did not derive values of the activation energy in this paper. However, they continued their study, and in 1998 [217] came out with an equation describing the change in the activation energy for diffusion as a function of antimony coverage Θ ED ¼ ED0 þ 1:7Y:

(5:1)

Movement of gold atoms on a Ag(111) surface was investigated by Haftel and Rosen [218] using molecular dynamics with EAM potentials. Hopping of Au on Ag(111) required an energy of 0.102 eV and showed a frequency prefactor of 1.3 × 1012 sec−1, evaluated in the harmonic approximation to the attempt frequency

321

5.14 Silver: Ag(111)

(a)

Ea ED

EB

(b) Sb

Sb Sb

ED

Fig. 5.54

EB

Potentials affecting interlayer surface transport. (a) Step edge barrier EB limits interlayer transport. (b) Higher diffusion potentials lower step-edge effects, stimulating layer-by-layer growth (after Vrijmoeth et al. [216]).

EðsÞ ¼ 1=2mω2d s2 :

(5:2)

Here E(s) gives the energy as a function of s, the displacement from the initial equilibrium, m is the mass of the atom diffusing, and ωd gives the angular attempt frequency. The frequency factor can also be estimated based on the number of diffusion events in the interval t 5n4ðtÞ ¼ Na νh

ðt dt exp½
ED =kTðtÞ;

(5:3)

0

where Na is the number of atoms simulated, νh is the harmonic aproximation attempt frequency of the diffusing atom. The frequency based on this equation was 1.19 × 1012 sec−1 and the barrier amounted to 0.102 eV, the same as in their previous derivation. For exchange the energy barrier was bigger, 0.421–0.805 eV, with a frequency factor of 1.5 × 1012 sec−1, which makes this event unlikely. However, an atom-catalyzed double exchange (already discussed in Chapter 3) on the terrace cost only 0.047–0.083 eV with frequency factors of 2.1 × 1012 sec−1 based on Eq. (5.2) and 1.9 × 1010 sec−1 based on Eq. (5.3). The diffusion of copper adatoms on Ag(111) was examined by Morgenstern et al. [219] using scanning tunneling microscopy. Atoms moved primarily over distances comparable with a lattice spacing. An energy difference of 0.0055 ± 0.0010 eV was found between fcc and hcp sites. A plot of the diffusivity against 1/T is given in Fig. 5.55, and yields a diffusion barrier of 0.065 ± 0.009 eV and a frequency prefactor of 1 × 1012 ± 0.5 sec−1 [220]. This is one of not many direct studies of adatom movement by STM. From molecular dynamics simulations using potentials from effective medium theory, but disregarding long-range interactions, they obtained a value of 0.08 eV.

322

Diffusion on two-dimensional surfaces

H

(a)

H L

R H R H

H

0:00:00

Diffusivity D (nm2/sec)

3:58:48

e–7

(b)

e–8 e–9 e–10 e–11 e–12

ED =0.065 ± 0.009 eV

e–13 450

500

550

600

1/KT (eV–1) Fig. 5.55

Copper adatom diffusivity on Ag(111) plane. (a) Time lapse STM images at 21.5 K. (b) Arrhenius plot of diffusivity, giving an activation energy of 0.065 ± 0.009 eV (after Morgenstern et al. [219]).

A study of the interactions between cerium adatoms on Ag(111) was done by Silly et al. [221] using the STM. In the course of this work they also measured the time at 4.8 K for a cerium atom to execute a jump to be 0.3 sec; assuming a frequency prefactor of 1012 sec−1 they arrived at a diffusion barrier of 0.011 eV for cerium on Ag(111). The molecular dynamics simulations based on RGL interactions of Kim et al. [70] recently yielded the following barriers: Ni 0.071 eV, Cu 0.078 eV, Pd 0.087 eV, Pt 0.040 eV, and Au 0.088 eV. The results for Cu and Au are not far away from previous studies. There clearly is not enough information available for hetero-diffusion on Ag(111) for any trends to emerge.

5.15

Silver: Ag(100)

5.15.1

Self-diffusion Investigations of the self-diffusion on Ag(100) started with the study of Venables in 1987 [200] who reanalyzed the data of Hartig et al. [222] for the island density of silver deposited on a Ag layer on top of a Mo(100) surface. Measurements were done at temperatures from 150 °C to 600 °C with scanning electron microscopy in combination

5.15 Silver: Ag(100)

323

with RHEED and AES. Venables derived a barrier of 0.45 eV for movement of silver on this surface. The effect of the possible influence of strain on atom movement was not disscused. Some studies of silver diffusing on Ag(100) were done by Langelaar et al. [223] in 1996. They irradiated the surface with 6 keV Ne+ ions, which create Ag adatoms on the (100) surface. As the surface temperature for irradiation increased from 160 K, the yield of silver adatoms was essentially constant. Langelaar et al. determined 160 K to be the temperature at which migration began. Assuming a value of the frequency prefactor of 1 × 1012 sec−1, they arrived at an activation energy of 0.40 ± 0.05 eV. To allow for an uncertainty of a factor of 10 in their assumed frequency prefactor, they increased the error estimate for the barrier by 0.03 eV. The irradiation in this study created vacancies as well as adatoms, and it is not clear how this may have affected adatom motion. Zhang et al. [224] also examined the homo-epitaxy in 0.1 monolayers of silver on Ag(100), studying nucleation by scanning tunneling microscopy. Comparing experiments with Monte Carlo simulations without atomic interactions, they arrived at an activation energy for surface diffusion of 0.330 ± 0.005 eV, assuming a prefactor ν0 = 1 × 1012 sec−1. Next year another study of this system by Zhang et al. [225] appeared. From an STM examination and a Monte Carlo simulation of the homo-epitaxy using a lattice gas model they arrived at a higher value of the activation energy of 0.38 eV, assuming a frequency prefactor ν0 = 1013 sec−1; this was subsequently revised up, to a barrier of  0.40 eV [226]. Bardotti et al. [227] did high resolution LEED measurements for the submonolayer growth of silver at 0.3 of a monolayer on Ag(100). From the splitting of the diffraction profiles over a range of temperatures from 170 K to 295 K, they were able to arrive at the mean island separation, which gave them an activation energy for silver diffusion ranging from 0.37 ± 0.06 eV to 0.45 ± 0.06 eV. Comparison of these results with previous STM studies led them to a value of 0.40 ± 0.04 eV for the diffusion barrier, assuming a frequency prefactor 3 × 1013 sec−1. Work with perturbed angular correlation of 111Ag atoms on Ag(100) was carried out by Rosu et al. [228]. From the terrace length distribution, and an estimate of 100 hops to reach a lattice step, they reached a diffusion barrier of 0.35 ± 0.03 eV assuming a frequency prefactor of 1012 ± 1 sec−1. For diffusion by atom exchange they estimated an activation energy of 0.45 ± 0.03 eV. In 1999 Laurens et al. [96] looked at the same system with the same technique. One hundredth of a monolayer of 111Ag was deposited on a Ag(100) surface. The adatoms were immobile below 120 K. Steps started to be populated at 130 K, indicating the onset temperature for diffusion and leading to an activation energy of 0.37 ± 0.03 eV. The barrier for exchange was measured as 0.48 ± 0.03 eV, all in agreement with previous findings. The low coverage was probably suitable to learn about single atom motion. The formation of two-dimensional silver islands in 0.1 monolayers of silver on a Ag(100) surface was followed by Frank et al. [229]. From STM studies of island densities shown in Fig. 5.56 they found a diffusion barrier of 0.40 eV and a frequency prefactor of 5 × 1012 sec−1. These results are close to the work of Langelaar et al. [223], but all except the last study are based on an assumed value for the prefactor, so a better

324

Diffusion on two-dimensional surfaces

180 K

250 K

200 K

225 K

280 K

300 K

Island Density (site–1)

10–1

200

Temperature T (K) 232 193 166 145

129

10–2

10–3

ED = 0.40 eV

ν0 = 5 × 1012 sec–1 10–4 (b)

40

50

60

70

80

90

1/kT (eV–1)

(a) Fig. 5.56

Deposition of Ag on Ag(100). (a) STM images of growth at different temperatures. Coverage 0.1 ML, deposition rate 0.006 ML/sec. (b) Arrhenius plot of island density yields an activation energy for diffusion of 0.40 eV (after Frank et al. [229]).

examination here would not be amiss. However, the experimental barrier heights for selfdiffusion on Ag(100) are all between ~0.3 eV and ~0.4 eV, which is a hopeful sign that they reflect adatom movement on this surface. Theoretical work for the diffusion of silver atoms on silver surfaces started with the work of Flahive and Graham [47], using Morse potentials to give a barrier of 0.35 eV as shown in Fig. 4.9. Voter [230] was among the first to make an estimate of the barrier to diffusion by hopping allowing for many-atom effects. Resorting to EAM potentials between atoms and allowing 43 surface atoms to be movable he arrived at a value of 0.49 eV for the activation barrier. For a vacancy the diffusion barrier was only 0.46 eV. Further work with EAM potentials was done by Liu et al. [4]. They arrived at an activation energy of 0.48 eV with AFW parameters yielding a prefactor of 1.2 × 10−3 cm2/sec, and the same barrier but with a prefactor of 3.9 × 10−3 cm2/sec using VC curves. For atom exchange they found higher barriers, 0.75 eV from AFW and 0.60 eV from VC with a prefactor of 2 × 10−2 cm2/sec. Sanders and DePristo [76] reported a barrier of 0.24 eV and a frequency prefactor of 1.0 × 1012 sec−1 for hopping with the MD/MC-CEM approximation, a value lower than previous findings. However, this changed with the next study in the same group. Perkins and DePristo [77] came up with an activation energy of 0.45 eV for hopping; the exchange mechanism seems to have been unfavorable on this surface with a high barrier

5.15 Silver: Ag(100)

325

of 1.01 eV. They also did CEM calculations to find a barrier of 0.41 eV for hopping and 0.58 eV for atom exchange. Two years later, Perkins and DePristo [29] investigated the influence the number of movables had on the value of the activation energy for hopping as well as exchange. Their results for silver were similar to what was found for other substrates: the barrier to hopping did not depend sensitively on the number of atoms movable, but for atom exchange it did very strongly. With calculations based on MD/ MC-CEM interactions, increasing movables from 13 to 226 lowered the barrier to exchange of silver from 1.01 eV to 0.70 eV, and with CEM interactions from 0.58 eV to 0.41 eV. It is clearly difficult to discern which process is favored. Nelson et al. [207] made estimates using EAM, but with FBD set Universal3 [57], and arrived at a barrier of 0.479 eV. The calculations, done on a static lattice, considered only hopping. Stoltze [61] relied on EMT and found an activation energy for self-diffusion of 0.365 eV. Cohen [7] carried out molecular dynamics simulations relying on EAM interactions, and identified a “new” diffusion process in which an adatom entered the substrate and a lattice atom subsequently emerged a few spacings away. The temperature dependence for these events, as well as for hopping and the usual exchange, were shown in the Arrhenius plot in Fig. 3.62. Exchange begins to be important above ~ 600 K, but long-range exchange only becomes significant at 1000 K. Full-potential linear-muffin-tin-orbital estimates were made by Boisvert et al. [208], who found an energy barrier for diffusion on Ag(100) of 0.50 ± 0.03 eV, higher than previously reported values. Continuation of this study with molecular dynamics simulations relying on the EAM [78] gave for hopping values of 0.48 eV and a prefactor of 1.53 × 1013 sec−1. For exchange they arrived at a much higher activation energy of 0.78 eV and a prefactor of 3.92 × 1014 sec−1. Yu and Scheffler [231,232] used plane-wave pseudo-potential calculations and density-functional methods in a 3 × 3 cell to arrive at an activation energy of 0.52 eV in the local-density approximation and 0.45 eV using the generalized gradient approach. For diffusion via the exchange mechanism they found a bigger barrier, 0.93 eV from LDA and 0.73 eV using GGA. Shi et al. [79] used the embedded atom method (EAM) to calculate the activation energy to be 0.478 eV. Effective medium theory was used by Merikoski et al. [81] to come up with a hopping barrier of 0.367 eV; for diffusion by atom exchange they found a higher barrier of 0.614 eV. Kürpick et al. [114] calculated a barrier for hopping of 0.48 eV with both EAM potentials, FBD and VC. For the system at 600 K, Kürpick and Rahman [80] reported a slight increase, to 0.49 eV using FBD potentials and 0.50 eV with potentials of type VC; prefactors were 3.4 × 10−4 cm2/sec (FBD) and 5.5 × 10−4 cm2/sec (VC). The next year Kürpick and Rahman [233] continued work with EAM potentials to calculate the same activation energy as above for atom hopping on Ag(100) with a prefactor of 8.1 × 10−4 cm2/sec for FBD potentials and 2.3 × 10−3 cm2/ sec with VC parameters in the harmonic aproximation. The result in the quasi-harmonic approximation at 600 K was as shown a year earlier [80], but with prefactors of 9.3 × 10−4 and 1.5 × 10−3 cm2/sec for FBD and VC potentials, respectively. No temperature dependence of the prefactor was detected. In 1998, Papanicolaou et al. [212] used molecular dynamics with a tight-binding RGL model and for bridge hopping they found an energy barrier of 0.43 ± 0.04 eV and

326

Diffusion on two-dimensional surfaces

Diffusivity D(cm2/sec)

10–4 Exchange ED = 0.59 ± 0.02 eV

Ag/Ag(100)

10–5

10–6

10–7 12

Double Exchange ED = 0.74 ± 0.06 eV

14

Hopping ED = 0.43 ± 0.04 eV 16

18

20

1/kT (eV–1) Fig. 5.57

Arrhenius plots for hopping and exchange frequencies of Ag on Ag(100), obtained from molecular dynamics simulations with RGL potentials (after Papanicolaou et al. [212]).

Do = (1.4 ± 0.6) × 10−3cm2/sec. Exchange cost 0.59 ± 0.02 eV, with a prefactor for the diffusivity of (40 ± 10) × 10−3cm2/sec, one range higher than for hopping; the static barrier for hopping amounted to 0.46 eV. Double exchange required an activation energy of 0.74 ± 0.06 eV, and was characterized by a prefactor of (300 ± 200) × 10−3 cm2/sec, ten times higher than for simple exchange and 100 times higher than for hopping. The exchange mechanism started to be important for this surface at a temperature around 600 K. Their data for the transition frequency at different temperatures are illustrated in Fig. 5.57. Calculations were done by Zhuang and Liu [119] using potentials from SEAM [234]. From rate measurements over a range of temperatures they arrived at an activation energy of 0.26 ± 0.04 eV, rather lower than the static barrier of 0.39 eV they derived for simple exchange. They concluded that diffusion involved other strain-induced exchanges. Mehl et al. [82] relied on EAM potentials to come up with a barrier of 0.48 eV; they considered only hopping in a 3 × 3 cell.With his cohesion approximation, Davydov [49] estimated a self-diffusion barrier of 0.06 eV on (100), unusually low. Baletto et al. [214] looked at diffusion on a silver Wulff polyhedron with the nudged elastic band method and RGL. On this polyhedron a (100) plane consisted of nine atoms. On such a plane they found that diffusion proceeded by hopping with a barrier of 0.43 eV, not really different from the barrier on a big, flat plane. Depending on Lennard-Jones potentials, Agrawal et al. [26] found the barrier for diffusion as 0.52 eV with a prefactor of 2.2 × 10−3 cm2/sec. Chang and Wei [28] estimated a hopping barrier of 0.60 eV using ab initio VASP simulations, and 0.76 eV for diffusion by atom exchange. The results obtained by Müller and Ibach [127] of 0.62 eV for the hopping barrier using a local-density approximation are close to this value. They also made an estimate of 1.00 eV for the barrier to self-diffusion by atom exchange. Kim et al. [70] have recently done molecular dynamics simulations with RGL interactions to find a barrier of 0.467 eV for atom jumps in self-diffusion, and 0.624 eV by exchange. The scatter in the barriers calculated for self-diffusion on Ag(100) is really not too bad, independent of the technique. The values for hopping range from 0.36 eV to 0.62 eV, excluding the results of Davydov [49], and correspond not too badly with the barriers

5.15 Silver: Ag(100)

327

from experimental studies, 0.3 eV to 0.4 eV. It is clear that many-body potentials are required to study this system. Atoms move by hopping rather than exchange on the Ag(100) plane, but a lot of calculations have not looked into exchange as a possible mechanism of diffusion. However, as shown by Perkins and DePristo [29], the number of substrate atoms considered as movable in simulations has a large effect on the barrier to exchange, and this factor should be more carefully considered. In any event, further experiments, especially direct motion investigations on Ag(100), would be quite desirable.

5.15.2

Hetero-diffusion Considerable work has been done on hetero-diffusion on the (100) plane of silver. Fink et al. [235,236] looked at diffusion of indium atoms on Ag(100) using perturbed angular correlation spectroscopy with 111In, at a low concentration of 10−4 monolayers. For the activation energy they arrived at a value of 0.31 ± 0.03 eV, assuming a frequency prefactor of 1012 sec−1. In 1996, Patthey et al. [237] examined diffusion of palladium atoms over the Ag(100) surface. The coverage was estimated to be 0.1 ML. From measurements of the time rate of change of the photoelectron spectra at 135 K they arrived at a barrier of 0.43 ± 0.05 eV for palladium diffusion on this plane. To reach this value it was assumed that the frequency factor is the same as at higher temperatures, where subsurface diffusion is presumed to occur. Using EAM potentials, Patthey et al. calculated an activation energy of 0.53 eV for exchange of palladium with a silver atom from the lattice, which compares favorably with the measured barrier. Based on this they assumed that movement proceeded by exchange rather than by hopping. Calculations for hetero-diffusion by hopping on Ag(100) were done by Sanders and DePristo [76]. Their work, based on MD/MC-CEM interactions and transition-state theory, yielded the following results: nickel barrier 0.71 eV, frequency prefactor 5.0 × 1012 sec−1; copper 0.73 eV, 5.6 × 1012 sec−1; rhodium 0.74 eV, 4.5 × 1012 sec−1; platinum 0.59 eV, 3.7 × 1012 sec−1; gold 0.54 eV, 3.2 × 1012 sec−1. For diffusion of palladium the barrier was 0.58 eV, with a prefactor of 5.0 × 1012 sec−1, a value for the activation energy somewhat higher than the 0.43 eV found in experiments by Patthey et al. [237]. These experiments suggested that atom exchange, not considered in these calculations, would be rather favorable. Two years later this was looked at by Perkins and DePristo [89], who did work with 81 movable surface atoms in the first layer. Their results for the diffusion barrier by hopping compared to exchange are given in Table 5.4. Prefactors were in the range 1012 sec−1, but the values for atom hopping were at least twice as large as for exchange and are listed in Table 5.4. The barrier for palladium atom exchange agreed quite nicely with the experiments. According to this study in hetero-diffusion, exchange was the favored mechanism on the Ag(100) surface, except for copper atoms; there the barriers for hopping and exchange were too close together to reach a decision. Canepa et al. [238] looked at the growth of iron on a Ag(100) surface with ion scattering spectroscopy. From observations of the alloying process they concluded that exchange diffusion would explain their findings, but this process was not checked in detail.

328

Diffusion on two-dimensional surfaces

Table 5.4 Diffusion on Ag(100) [89]. Frequency Prefactor (×1012 sec−1) Diffusion Barrier (eV) Adatom Bridge Hops

Exchange

Hopping

Exchange

Rh Ni Pd Cu

3.43 7.29 4.01 14.44

0.73 0.70 0.76 0.55

0.26 0.33 0.38 0.52

9.69 18.24 16.08 34.02

In 2005 Caffio et al. [239] examined the growth of Ni layers on Ag(100) with a number of techniques, LEIS, XPS, STM, and XPD. To understand their findings they employed EAM simulations with a 13 × 13 unit cell and a slab of 5 layers; for hopping of nickel on the Ag(100) surface they found a barrier of 0.61 eV, big compared with the barrier of 0.19 eV for exchange. Their barriers are lower then the findings of Perkins and DePristo [89], but the difference between hopping and exchange is in the same range. The movement of an Ag atom by exchange away from an embedded Ni atom at the surface cost 0.59 eV. Molecular dynamics simulations by Kim et al. [70], based on RGL potentials, yielded the following barriers for hetero-atom hopping on Ag(100): Ni 0.529 eV, Cu 0.569 eV, Pd 0.674 eV, Pt 1.189 eV, and Au 0.662 eV. Unfortunately, no estimates for exchange were given. The especially high value of the activation energy for Pt atoms suggests that a different mechanism might be in order. However, such a high energy barrier was not seen previously by Sanders and DePristo [76]. Not enough information is available about hetero-atom diffusion on Ag(100) to discern any trends.

5.16

Tantalum: Ta(110) Only brief measurements are available for the diffusion of palladium on the Ta(110) surface. Based on only forty-five observations in an FIM at each of three temperatures, Schwoebel and Kellogg [240] arrived at an estimate of 0.49 ± 0.02 eV for the barrier to hopping, assuming a prefactor of 2 × 10−3 cm2/sec. Davydov [49], with his cohesion approximation, came up with a self-diffusion barrier of 1.19 eVon Ta(110); like many of his results, this study is likely to not predict diffusion correctly. Antczak and Blaszczyszyn [241] looked at the spreading of palladium at two monolayers over a tantalum tip in field emission experiments. They reported an activation energy of 1.45 eV, but this is likely to be the energy of moving over vicinals of the (110) plane rather than over the smooth surface. Furthermore, the layer was really too concentrated to derive single atom information. Using Morse potentials, Flahive and Graham [47] estimated a barrier of 2.2 eV for selfdiffusion on Ta(111), 1.0 eVon Ta(100), and 0.45 eVon Ta(110), as indicated in Fig. 4.9. The last value is in good agreement with experiments.

329

5.17 Tungsten: W(110)

There is almost a complete lack of information about diffusion on this surface, and more work clearly needs to be done.

5.17

Tungsten: W(110)

5.17.1

Self-diffusion experiments The first ever measurements of single atom self-diffusion [242] were done in field ion microscopic experiments on the (110) plane of tungsten in 1966. An Arrhenius plot of these measurements is given in Fig. 5.58. Unfortunately the measurements were done with a few atoms on the plane and later on it was demonstrated that the other atoms on the surface could influence the movement observed. Ayrault improved the data analysis [167] and came up with a value of 0.92 ± 0.05 eV for the diffusion barrier, with a prefactor of 2.6 × 10−3 cm2/sec, which agrees nicely with the value accepted nowadays. A few years later, Bassett and Parsley [243] extended these studies, but still with a few atoms present on the plane, again using field ion microscopy. They determined an activation energy equal to 0.86 eV, with a prefactor Do = 2.1 × 10−3 cm2/sec. More introductory experiments, but with one atom on the plane were done by Cowan and Tsong [244], who reported a diffusion barrier of 0.77 ± 0.06 eV and a value of Do = (4.5 ± 0.2) × 10−4 cm2/ sec. Both the activation energies and the prefactors are somewhat low in these measurements. Kellogg et al. [245] in 1978 corrected these results for an error in the temperature calibration and arrived at an activation energy for diffusion of 0.90 ± 0.07 eV as well as a prefactor of 6.2(×13±1) × 10−3cm2/sec.

Mean-square displacement (Å2) per sec N02/t

10

Temperature T(K) 350 300

(110) 1.0

0.1

0.01

0.001 2.8

Fig. 5.58

Ayrault: ED = 0.92 ± 0.05 eV Do = 2.6 × 10–3 cm2/sec 3.0 3.2 3.4 1000/T (K–1)

3.6

Arrhenius plot for diffusivity of tungsten adatom on W(110) surface [242]. Barrier eventually determined as 0.92 ± 0.05 eV [167].

330

Diffusion on two-dimensional surfaces

3

W on W(110)

In [100 K. (b) Saturation island density at 0.12 ML coverage as a function of reciprocal temperature. For critical island size i = 1, diffusion barrier is 0.168 ± 0.005 eV (after Brune et al. [93]).

barrier a value of as 0.20 eV, reasonably close to the experiments. In his calculations, fcc sites are 0.03 eV more favorable than hcp sites. Ratsch et al. [210] also evaluated the activation energy for silver diffusion on Pt(111) using density-functional theory, and arrived at a value of 0.15 eV. There is quite good agreement of the experimental data with this study of Ratsch et al. [207]. Utilizing EAM interactions, Leonardelli et al. [330] looked at nickel on Pt(111), for which they found no metastable position similar to what they observed in self-diffusion. The activation energy for nickel diffusion was 0.178 eV. Nickel was also investigated by Kim et al. [70] with molecular dynamics simulations and RGL interactions. They derived a value of 0.114 eV. Unfortunately there is no experimental data for comparison. In 2003, Sabiryanov et al. [339] looked at the influence of strain on the movement of Co on Pt(111) using ab-initio calculations [340] and the hopping barrier on the unstrained surface was estimated as 0.3 eV. Bulou and Massobrio [68] in 2005 used a second-moment approximation of tight-binding to calculate an activation energy for exchange of cobalt on Pt(111) as 2.065 eV. Long-range exchange required a higher energy of 2.248 eV, indicating that Co adatoms will be moving by jumps, not by exchange. Goyhenex [341] also employed the second-moment approximation to the tight-binding formalism to come up with an activation energy of 0.21 eV for cobalt atoms diffusing on Pt(111); the energy difference between fcc and hcp sites was negligible, less than 0.01 eV. From the calculated mean-square displacement at different temperatures and an Arrhenius plot, shown in Fig. 5.93, he derived a diffusion barrier of 0.23 ± 0.01 eV and a prefactor Do = 3.9 × 10−5 cm2/sec. It appears that jumping is the mechanism of movement for Co on the Pt(111) surface; exchange is very unlikely due to the huge energy required. Graham and Toennies [342] looked at the diffusion of sodium on a Pt(111) surface with quasielastic helium atom scattering. They found that sodium at a coverage of

361

5.24 Platinum: Pt(100)

–14 –15

In D

–16 –17 –18 –19

ED = 0.23 ± 0.01eV

–20

Do = 3.9 × 10–5 cm2/sec 20

25

30

35

40

1/kT (eV–1) Fig. 5.93

Arrhenius plot for the diffusivity of Co atoms on Pt(111), leading to a diffusion barrier of 0.23 eV. Data obtained by molecular dynamics simulations (after Goyhenex [341]).

0.043 ML moved over the surface at a low temperature of 199 K, with an effective activation energy from an Arrhenius plot of 0.0215 ± 0.001 eV and a prefactor (4.5 ± 0.3) × 10−4cm2/sec. The true diffusion barrier was estimated to be in the range of 0.030 to 0.044 eV, with Do = 6 × 10−3 cm2/sec. The coverage in this study should be suitable for an investigation of single atom movement. Estimates for hetero-atom diffusion on Pt(111) were made by Kim et al. [70] from molecular dynamics simulations based on RGL interactions. The barriers were as follows: Ni 0.114 eV, Cu 0.111 eV, Pd 0.128 eV, Ag 0.090 eV, and Au 0.122 eV. Only for Ag on Pt(111) is there enough data for comparisons. The experimental results look reasonable and are close to the density-functional estimates of Ratsch et al. [210].

5.24

Platinum: Pt(100) The motivation for work on Pt(100) was provided by DeLorenzi and Jacucci [3] in 1985, who did molecular dynamics simulations of diffusion phenomena. They simulated atomic behavior using a potential derived by Price [343]. On a bcc(211) surface they observed primarily diffusion along the channels, with occasional exchange events involving a lattice atom from the row. On (100), motion by single jumps was observed, but most interesting were more complicated transitions, indicated in Fig. 3.52. In these, an adatom enters the surface layer, replacing an atom from the lattice, which moves to the surface. These events, previously modeled in Fig. 3.53, provided the stimulus for closer examination of diffusion on (100) surfaces.

5.24.1

Self-diffusion In 1990, Kellogg and Feibelman [344] finally examined self-diffusion on Pt(100) using the field ion microscope. Atoms were found to move diagonally, creating a c(2 × 2) net of sites such as shown in Fig. 3.54, which indicates atom exchange. From the mean-square

362

Diffusion on two-dimensional surfaces

displacement at 175 K, Kellogg and Feibelman estimated an activation energy of 0.47 eV for diffusion, assuming a prefactor of 1 × 10−3 cm2/sec. Kellogg [345] has further looked at diffusion on the (100) plane of platinum. From an Arrhenius plot of the diffusivity, in Fig. 3.56, he arrived at an activation energy of 0.47 ± 0.01 eV and a prefactor Do = 1.3(×10±1) × 10−3 cm2/sec for self-diffusion. A few years later, in 1994, Kellogg [346,347] looked at the effect of an electric field on the diffusion of Pt atoms on Pt(100), which occurs by atom exchange. These introductory studies showed an interesting effect. A negative field (that is with a negative voltage on the emitter) increased the diffusivity, a positive field diminished it. Quite different is the behavior on Pt(311), on which self-diffusion is in-channel and occurs by hopping. Here the field has really no effect. At higher positive fields on Pt(100), the activation energy for diffusion continued to rise and at 1.5 V/Å the map of sites at which atoms are seen, shown in Fig. 5.94, changed from c(2 × 2) to (1 × 1), suggesting a change from atom exchange to hopping. The scanning tunneling microscope was used by Linderoth et al. [348] to measure the saturation density of platinum islands at a coverage of 0.07 ML on the reconstructed Pt(100)-hex surface in Fig. 5.95. Diffusion on this plane was highly anisotropic. They concluded that dimer mobility was negligible in the investigated range of temperatures

Mean-Square Displacement (Å)

250 Pt on Pt(001) T = 190 K Exchange

200 150 100 50 0

(a) Pt on Pt(311) at 222 K Hopping

–0.5

[100] Fig. 5.94

T = 200 K

(c)

0.5

Fe = 1.75 V/Å

T = 265 K

[010]

Fe = 0 V/Å

[010]

(b)

0.0 Applied Electric Field

[100]

Effect of electric field on diffusion of platinum on Pt(001) plane, observed in FIM. (a) Mean-square displacement of Pt in its dependence upon the applied electric field. (b) Sites filled after diffusion in zero field indicate movement by exchange with a lattice atom. (c) In electric field and at higher temperature, distribution suggests single atom hops (after Kellogg [346,347]).

363

5.24 Platinum: Pt(100)

(a)

(b)

nx(site–1)

(c) 10–3

10–5 –13 10

Fig. 5.95

318 K 357 K 389 K 439 K 497 K

10–4

10–12 10–11 10–10 R/ν exp(–ED /kT )

10–9

STM images of Pt(100)-hex surface. (a) (100) surface. (b) After Pt atom deposition at 389 K. Inset: resolved edge of island. (c) Platinum island density measured at different temperatures (after Linderoth et al. [348]).

and used Monte Carlo simulations to explain their measurements, arriving at a diffusion barrier of 0.43 ± 0.03 eV. In 1998, Mortensen et al. [349] continued the study of Linderoth et al. [348] with an STM examination of platinum deposition on Pt(100)hex. They found that diffusion along the six-atom wide channels dominated. Only a small number of theoretical estimates has been made for diffusion on Pt(100), starting with the investigation by Flahive and Graham [47] using Morse potentials, shown in Fig. 4.9; they calculated a barrier of 0.8 eV. Early calculations were done by Liu et al. [4] with EAM interatomic potentials. With parametrization according to AFW they found an activation energy of 0.44 eV for atom jumps and a prefactor of 8 × 10−4 cm2/sec. With VC parameters, they arrived at a much higher barrier of 1.25 eV, which also gave a higher prefactor of 5.0 × 10−3 cm2/sec. Atom exchange occurred over a smaller activation energy of 0.31 eV with AFW parameters, and 0.64 with VC potentials, which gave a prefactor of 1 × 10−2 cm2/sec. It is clear that exchange not hopping will dominate movement. Using Lennard-Jones potentials, hopping required overcoming a barrier of 1.05 eVand involved a prefactor of 4.9 × 10−3 cm2/sec; with Morse interactions the barrier was lower, 0.82 eV. For atom exchange, Lennard-Jones interactions yielded a huge barrier of 3.97 eVand a prefactor of 4 × 10−2 cm2/sec, showing that the L-J potential is not suitable for modeling processes on platinum surfaces.

364

Diffusion on two-dimensional surfaces

Roughly at the same time Lynden-Bell [176] relied on molecular dynamics calculations for self-diffusion of platinum using the Sutton–Chen potential. She found a barrier of 0.611 ± 0.004 eV and a prefactor of 3.5(×0.2±1)×10−2 cm2/sec for atom exchange. Sanders and DePristo [76] did transition-state theory estimates of atom hopping based on MD/MC-CEM interactions and reported an activation energy of 0.72 eV as well as a frequency prefactor of 2.5 × 1012 sec−1. For hopping of platinum atoms, Stoltze [61], using effective medium interactions, estimated a barrier of 0.689 eV. In 1994 Villarba and Jónsson [323] used EAM potentials to obtain a barrier of 0.54 eV for atom exchange on the Pt(100) surface. Calculations by density-functional theory were carried out by Feibelman and Stumpf [179] in 1999. For diffusion by atoms jumping over the surface the activation energy proved to be 1.04 eV. For diffusion by atom exchange with the substrate the barrier was only 0.38 eV, clearly the lower energy process, but lower than the experimental values. Davydov [49], with his cohesion based calculations, reported a barrier of only 0.07 eV for platinum jumps over this surface, definitely too low and out of range. Estimates were done by Zhuang and Liu [350] who derived an activation energy for exchange of 0.989 eV from conventional EAM and 0.865 eV using the potential of Oh and Johnson [58]. Zhuang and Liu [119] continued with the surface embedded atom method [234,351,352]. They arrived again at an activation energy of 0.98 ± 0.04 eV, in very good agreement with their static value of 0.99 eV to simple exchange, but much higher than the experimental value. For hopping they found a barrier of 1.84 eV. Further work by Zhuang and Liu [353] came up with a new mechanism for self-diffusion on Pt(100). In this process, labeled ad-dimer diffusion, an adatom moving into the lattice displaces two atoms from their sites, of which the second one remains on the surface as an adatom. Zhuang and Liu estimated a barrier 0.02 eV higher for conventional exchange than for ad-dimer diffusion, which should certainly participate in migration. Estimates were made by Agrawal et al. [26] with Lennard-Jones potentials for atomic interactions, and once more show the importance of many-body effects in modeling platinum surfaces. They gave a barrier height of 1.02 eV and a prefactor of 2.1 × 10−3 cm2/sec for atom jumps on this surface. In 2004, Xiao et al. [354] looked at a strained Pt(100) surface using the embedded atom method. They first examined exchange in a (5 × 5) cell and found an activation energy of 0.33 ± 0.05 eV. When they increased the cell to (10 × 20), the activation energy decreased to 0.27 ± 0.05 eV; a further increase of the cell size did not lower the energy. For hopping they found a barrier of ~0.56 eV±0.05 eV for a (5 × 5) cell and 0.36 ± 0.05 eV for a 10 × 20 cell. Chang and Wei [28] in 2005 arrived at a very high energy of 1.2 eV for atom hopping on Pt(100) using the ab initio VASP package [22–24]. For diffusion by atom exchange the barrier was quite low, 0.31 eV. Kim et al. [70] found a barrier of 0.875 eV for Pt jumps using molecular dynamics estimates based on RGL potentials. For atom exchange, the barrier dropped to 0.388 eV, values which are again different from previous work. The barriers calculated for self-diffusion over Pt(100) are all over the map, but the original work of Liu [4] with AFW-EAM potentials came quite close to the experiments.

5.25 Gold: Au(100)

5.24.2

365

Hetero-diffusion For atoms other than platinum the experimental situation is not as good, since detailed measurements over a range of temperatures were not always possible. Kellogg et al. [355], however, from observations at one temperature of the diffusion of palladium deduced an activation energy of 0.70 ± 0.01 eV assuming a prefactor of 10−3 cm2/sec. Note the much higher activation energy for diffusion of palladium compared to platinum, which is undoubtedly due to the difference in diffusion mechanisms – palladium diffused by hopping, as can be seen from the site map shown in Fig. 3.57. Kellogg looked briefly at nickel as well and found atom exchange occurring with the lattice. Sanders and DePristo [76] calculated hetero-diffusion characteristics for atom hopping on Pt(100), and obtained the following results: nickel diffusion barrier 0.57 eV, prefactor 4.5 × 1012 sec−1; copper 0.61 eV, 4.5 × 1012 sec−1; rhodium 0.57 eV, 3.3 × 1012 sec−1; palladium 0.49 eV, 3.2 × 1012 sec−1; silver 0.41 eV, 9.3 × 1012 sec−1; gold 0.58 eV, 2.1 × 1012 sec−1. Hetero-atom diffusion barriers have also been given by Kim et al. [70], based on molecular dynamics simulations and RGL potentials. The barriers were 0.686 eV for Ni, 0.615 eV for Cu, 0.627 eV for Pd, 0.508 eV for Ag, and 0.507 eV for Au. No exchange was investigated in this study. The energy barrier for palladium lies somewhat closer to experiment than the findings of Sanders and DePristo [76]. The amount of information available about hetero-diffusion on Pt(100) is limited. The mechanism of Pd and Ni diffusion is clear, but other adatoms need further investigations. We note that the theoretical estimates did not consider the exchange mechanism at all.

5.25

Gold: Au(100)

5.25.1

Self-diffusion The first studies of the self-diffusion for gold were done by Günther et al. [356] on the reconstructed Au(100)-hex surface. They relied on scanning tunneling microscopic examination of growth and determined the island density at saturation when 0.2 monolayers were deposited at different temperatures, as shown in Fig. 5.96. Interpretation of the measurements was not simple, as there were doubts about the critical island size, and diffusion was highly anisotropic. Their final decision was that the critical size at low temperatures was equal to three, yielding a diffusion barrier of  0.2 eV. Liu et al. [357] offered a different set of assumptions to interpret the data of Günther et al.; using Monte Carlo simulations they arrived at a value of 0.35 eV for the barrier to atomic diffusion and 0.45 eV for the dimer, which gave reasonable agreement with experiments. Obviously the situation is not clear. Göbel and von Blanckenhagen [358] then did single surface scratch studies, measuring the width with an AFM, and came up with a diffusion barrier of 0.4 eVand a prefactor of 5 × 10−8 cm2/sec. This work was done in air on a gold sphere and diffusion does not yield information about the movement of individual atoms. In any event, the value of the prefactor is so low that these results are in serious doubt.

366

Diffusion on two-dimensional surfaces

Island Density (cm–2)

1012

Temperature T (K) 400 370 340

310

1011

Θ = 0.2 ML ED = 0.2 eV Deposition rate 0.5 ML/min 1010

Fig. 5.96

430

2.4

2.6 2.8 1000/T (K–1)

3.0

3.2

Mean Au island density observed with STM on hex-reconstructed Au(100) surface as a function of the reciprocal temperature (after Günther et al. [356]).

It appears that at the moment there is no really reliable value from experiments for the barrier to diffusion of gold atoms on any gold surface. In 1980 Flahive and Graham [47] used Morse potentials to estimate the barrier as 0.6 eV, as shown in Fig. 4.9. There have, however, been quite a number of more recent theoretical efforts to estimate diffusivities on gold surfaces. Liu et al. [4], with EAM atomic potentials, calculated a barrier to atom hops of 0.64 eV and a prefactor Do = 8 × 10−4 cm2/sec using AFW parameters; with VC potentials the barrier was even higher, 0.84 eV, and the prefactor amounted to 3.5 × 10−3 cm2/sec. For diffusion by atom exchange the barrier with AFW potentials turned out to be 0.30 eV, and for VC parameters 0.32 eV, with a prefactor of 1 × 10−2 cm2/sec. This suggests the exchange mechanism as the leading one, but this might change due to the reconstruction of this surface. The next calculations were done by Sanders and DePristo [76] using MD/MC-CEM for estimating interactions; they obtained a diffusion barrier for jumps of 0.67 eV and a frequency prefactor of 2.2 × 1012 sec−1 using transition-state theory. Stoltze [61], with effective medium theory potentials, looked at gold jumps and found activation energies of 0.49 eV. Molecular dynamics simulations, based on embedded atom interactions, were done by Boisvert et al. [78]. For jumps the activation energy proved to be 0.43 eV with a prefactor of 2.11 × 1013 sec−1, and for diffusion by atom exchange the numbers were 0.25 eV and 9.78 × 1012 sec−1. In this instance, diffusion clearly occurred by atom exchange. This study was continued with the full-potential linear-muffin-tin-orbital method by Boisvert et al. [208] to find a barrier of 0.62 ± 0.04 eV. Only hopping was considered in this study. Further study in the same group with molecular dynamics simulations and EAM [359] derived for atom jumps an activation energy of 0.49 eV, and a prefactor 1.56 × 10−2 cm2/ sec; from TST the barrier was essentially the same, 0.50 eV. The barrier for atom exchange proved to be much smaller, 0.26 eV with a prefactor of 5.8 × 10−3 cm2/sec, and 0.28 eV in TST estimates. Diffusion of gold atoms on the hexagonally reconstructed Au(100) surface was probed by Bönig et al. [360], who resorted to EMT potentials for the interactions. The atom paths in both the x- and y-directions on this surface are indicated in Fig. 5.97a. Total energy

367

5.25 Gold: Au(100)

top layer second layer

y

(a) x

–2667.7

–2667.7

(c)

(b)

–2668.0 –2668.1 10 Fig. 5.97

Energy

0.17 eV

Energy

–2667.9

0.28 eV

–2667.8

–2667.8

–2667.9 –2668.0

20

30 x

40

50

–2668.1

0

10

20

30

40

50

y

Diffusion of Au on Au(100). (a) Paths for diffusion on reconstructed (100) along both x- and y-direction. Total energy obtained with EMT potentials in diffusion of adatom along (b) x-direction, (c) y-direction (after Bönig et al. [360]).

estimates as an adatom moves in the x-direction are plotted in Fig. 5.97b. Also shown, in Fig. 5.97c, is the variation of the energy along the y-axis. For diffusion along the x-axis the barrier was between 0.15 eV and 0.17 eV. The more complex diffusion along the y-axis has as its largest barrier the value of 0.28 eV. In 1997 Yu and Scheffler [361] used density-functional theory to explore the possibility of exchange on fcc(100) planes. For self-diffusion on Au(100) they found that exchange is clearly favorable with an activation energy of 0.65 eV from the localdensity approximation [25] and 0.40 eV from the generalized gradient approach (GGA), using a (3 × 3) cell. This compares with a barrier for hopping of 0.83 eV from LDA and 0.58 eV from GGA. Calculations were done by Mehl et al. [82] with EAM potentials using a 3 × 3 cell; they came up with a diffusion barrier of 0.70 eV for hopping on the surface. Davydov [49] also made estimates of gold diffusion in his cohesion approximation; it proved to be negative, an unrealistic −0.03 eV. In 2000, Baletto et al. [214] looked at the movement of single atoms of gold on gold Wulff polyhedra using RGL potentials. On a 201 atom Wulff polyhedron, consisting of 9 atoms for the (100) facet, the atom moved by exchange with an activation energy of 0.41 eV, whereas the hopping barrier was 0.53 eV. On a 1289 atom Wulff-cluster the activation energy for

368

Diffusion on two-dimensional surfaces

jumps was slightly higher, on the order of a few hundredths of an electron volt. Agrawal et al. [26], relying on Lennard-Jones estimates, evaluated a barrier of 0.64 eV with Do = 1.5 × 10−3 cm2/sec. Chang and Wei [28] in 2005 resorted to density-functional theory and found a high barrier of 0.86 eV for atom hopping. For atom exchange the activation energy was only 0.32 eV. Müller and Ibach [127] did calculations using the local-density approximation to arrive at an activation energy of 0.64 eV. Estimates of the barrier to diffusion by atom exchange were also made, and yielded a similar result, 0.60 eV. Kim et al. [70] calculated a barrier of 0.531 eV, compared with 0.388 eV for atom exchange. Right now, results for self-diffusion of gold are not in good shape – there is really no convincing experimental work, and theoretical estimates have yielded widely spread values. At the same time gold (100) surfaces are likely to reconstruct, making comparison of experiment and theory more chellanging. Most theoretical work predicts exchange rather than hopping as the mechanism of movement, although some do not consider exchange at all.

5.25.2

Hetero-diffusion In 1993 He and Wang [362] examined the room temperature growth of a submonolayer of Fe on Au(100). From HRLEED and AES they claimed to see an atomic exchange of Fe with Au. No diffusion analyses were performed. Another indication of the exchange of iron atoms with Au(100) at room temperature was derivered by Herna´n et al. [363] using STM and MC simulations, but again movement was not characterized energetically. Sanders and DePristo [76] using MD/MC-CEM also examined hetero-diffusion on Au(100) and found the following characteristics: nickel barrier 0.61 eV, prefactor 4.6 × 1012 sec−1; copper 0.66 eV, 5.0 × 1012 sec−1; rhodium 0.62 eV, 3.6 × 1012 sec−1; palladium 0.53 eV, 3.7 × 1012 sec−1; silver 0.32 eV, 5.8 × 1012 sec−1; platinum 0.82 eV, 2.5 × 1012 sec−1. All of these estimates were for atom hopping on an unreconstructed surface, and without regard for the possibility of diffusion by atom exchange. Estimates of hopping barriers for hetero-diffusion were also made by Kim et al. [70], who relied on molecular dynamics simulations using RGL interactions on Au(100). They found 0.591 eV for diffusion of Ni, 0.547 eV for Cu, 0.582 eV for Pd, 0.437 eV for Ag, and 0.994 eV for Pt, all for atom jumps. Experiments, but also the many calculations, leave the activation energies for diffusion on gold in some doubt, and more definitive studies are clearly in order.

5.26

Gold: Au(111)

5.26.1

Self-diffusion There is no experimental information about the diffusion of single atoms on the Au(111) surface. This surface reconstructs, making the analysis more difficult. There are, however, quite a few theoretical determinations available.

369

5.26 Gold: Au(111)

In 1980 Flahive and Graham [47] used Morse potentials, as shown in Fig. 4.9, and derived a value of ~ 0.05 eV for the diffusion barrier on Au(111). There have been quite a number of more recent theoretical efforts to estimate diffusivities as well. These started with the work of Liu et al. [4], using EAM atomic potentials. A diffusion barrier of 0.021 eV was estimated using AFW potentials, with a prefactor of 2 × 10−4 cm2/sec; with VC potentials they found a barrier of 0.038 eV and a prefactor 3.7 × 10−4 cm2/sec. Stoltze [61], with effective medium theory potentials, found an activation energy of 0.102 eV. Ferrando and Tréglia [209] relied on RGL potentials and molecular dynamics simulations to obtain an activation barrier of 0.12 eV and a prefactor of 3.4 × 10−4 cm2/sec, shown in Fig. 5.52; the binding energy difference between fcc and hcp positions was equal to 0.005 eV. Molecular dynamics simulations, based on embedded atom interactions, were done by Boisvert et al. [78] and for atom jumps they found a barrier of 0.013 eV with a prefactor of 0.698 × 1012 sec−1. The full-potential linear-muffin-tinorbital method was also carried out by Boisvert et al. [208] to find a self-diffusion barrier of 0.22 ± 0.03 eV. Boisvert and Lewis [359], resorting again to EAM potentials made molecular dynamics studies, as shown in Fig. 5.98. The activation energy for the atomic jumps proved to be 0.015 eV, with a prefactor of 7 × 10−5 cm2/sec, in good agreement with the diffusivity as well as with transition-state theory, for which the barriers were 0.014 eV and 0.016 eV. Density-functional calculations were made by Mortensen et al. [63] in the localdensity approximation. They arrived at a barrier of 0.20 eV; in the generalized gradient method the barrier proved to be somewhat lower, 0.15 eV, but still much higher than in previous studies. In the same year Li and DePristo [62] used corrected medium theory to understand homoepitaxial growth on fcc(111) surfaces. For movement of a single gold atom on the gold (111) plane they found an activation energy of 0.029 eV.

D(Å/psec) or Γ(THz)

101 Γ (MD) D (MD) Γ (TST)

100

10–1

10–2 Au(111) 10–3

0

100

200

300

400

1/kT (eV–1) Fig. 5.98

Dependence of the diffusivity of Au adatoms on Au(111) upon the reciprocal temperature, calculated with EAM interactions. Diffusion barriers: from jump rate 0.015 eV, from diffusivity 0.014 eV, from TST 0.016 eV (after Boisvert and Lewis [359]).

370

Diffusion on two-dimensional surfaces

Davydov [49] also made estimates of gold diffusion in his cohesion approximation and surprisingly found a barrier equal to zero. In 2000, Baletto et al. [214] looked at the movement of single atoms of gold on gold Wulff polyhedra using RGL potentials. The jump from hcp to fcc site required 0.11 eV. With embedded atom potentials Chang et al. [35] in the same year estimated a barrier of 0.04 eV. Agrawal et al. [26], relying on Lennard-Jones estimates, evaluated a barrier height of 0.12 eVpand ffiffiffi a prefactor of 8.5 × 10−5 cm2/sec. In 2002 Liu [364] examined the strained (23 × 3) Au(111) surface. Using the glue potential [365,366], he observed anisotropic movement on such a plane with the diffusion barrier changing from 0.12 to 0.8 eV for different positions. Bulou and Massobrio [68] found a barrier to hopping of 0.112 eV and a much higher value, 0.878 eV for exchange, again using the second-moment approximation to tightbinding. Kim et al. [70] calculated a barrier of 0.117 eV for hops and 0.861 eV for exchange. On Au(111), investigators have rarely seen a difference between fcc and hcp sites, and it is clear that movement should proceed by hopping. The estimated activation energies are rather spread out and movement on the reconstructed surface has rarely been investigated.

5.26.2

Hetero-diffusion

pffiffiffi Diffusion of aluminum atoms on the (111)-(22 × 3) plane of gold has been examined by Fischer et al. [367]; they determined the island density at a coverage of 0.1–0.15 ML at different temperatures and applied kinetic Monte Carlo simulations. Based on their Arrhenius plot they concluded that the critical size equaled one for temperatures lower than 230 K. In the temperature range from 100 to 200 K, shown in Fig. 5.99, they found an activation energy of 0.030 ± 0.005 eV for diffusion and a prefactor ν0 = 7 × 103 ± 1 sec−1. The small value of the activation energy, but particularly the prefactor, casts doubt on this result, and raises concerns about the possible role of impurities and interactions on these findings. For temperatures higher than 245 K their distributions were bimodal and agreed with a critical nucleus of size i = 0, a sign of atomic exchange. Aluminum atoms sit on the Au(111) reconstruction–corrugation line, but there is no preferred place on such a line. Ovesson et al. [19] used density-functional theory calculations with GGA, and for Al on the Au(111) reconstructed surface derived a barrier of 0.12 eV and a frequency prefactor of 7 × 1012 sec−1. Meyer et al. [368] looked at the preferential adsorption sites for Ni atoms on a Au(111) herringbone-reconstructed surface, and explained the preferred sites by exchange of Ni atoms at the elbows. No energetics for thispprocess were derived. ffiffiffi Adsorption of cobalt on Au(111)-(22 × 3), also named as hex, was investigated by Goyhenex et al. [369] using tight-binding quenched molecular dynamics simulations. They derived the activation energy for diffusion which was measured as the difference between maximum and minimum in adsorption energies in adjacent places. For the discommensuration lines this difference was 0.18 eV, while in the fcc region 0.12 eV and in hcp 0.1 eV. This probably caused the slower diffusion across discommensuration lines with respect to other parts of the surface. Cobalt atom diffusion on unreconstructed

371

5.26 Gold: Au(111)

(a) T = 150 K

(b) T = 200 K

300 Å

300 Å (d) T = 250 K

(c) T = 235 K

600 Å (e) T = 260 K

600 Å (f) Temperature T(K) 245 200 150 120 100

10–2

10–3 nx 10–4

600 Å Fig. 5.99

3

4 5

10–5 6 7 8 9 10 11 1000/T (1/K)

Al films on Au(111), 0.1 ML coverage observed with STM. (a)–(e) Films deposited at different temperatures, at an impingement of 3.1 × 10−4 ML/sec. (f) Saturation Al island density as a function of reciprocal temperature. Diffusion barrier up to 200 K − 0.030 ± 0.005 eV (after Fischer et al. [367]).

Au(111) was tackled by Bulou et al. [370] in the second-moment approximation to tightbinding. From the mean-square displacement at different temperatures they were able to come up with an Arrhenius plot, shown in Fig. 5.100, which yielded an activation energy of 0.160 ± 0.003 eV and a prefactor Do = (5.8 ± 0.9)×10−4 cm2/sec. Static energy estimates gave a barrier of 0.155 eV for jumps from hcp to fcc sites and 0.137 eV for jumps in the opposite direction. Up from 810 K exchange started to be observed, with an activation energy of 1.1 eV. Two years later, Bulou and Massobrio [68] found an activation energy

372

Diffusion on two-dimensional surfaces

–9 –10 –11 In D

–12 –13 –14 –15

ED = 0.160 ± 0.003 eV –16 D = (5.8 ± 0.9) × 10–4 cm2/sec o –17 15 20 25 30 35 40 45 50 55 60 1/kT (eV–1) Fig. 5.100

Dependence of calculated Co diffusivity on Au(111) upon reciprocal temperature (after Bulou et al. [370]).

of 0.536 eV for atom exchange of cobalt on non-reconstructed Au(111) using a secondmoment approximation to tight-binding theory. They [371] also looked at the behavior of Co atoms on the Au(111) herringbone-reconstructed surface, and realized that the longest displacements occurred for Co atoms which started on fcc or hcp areas and ended in discommensuration line areas (DL). Cobalt atoms in DL areas moved along lines with smaller displacements than in fcc and hcp areas. They also observed an exchange mechanism at 600 K, 47.5% on DL areas, 47.5 % in hcp areas and 5% in fcc areas. Estimates of hetero-diffusion barriers were also made by Kim et al. [70], who did molecular dynamics simulations using RGL interactions. They estimated for atom jumps values for hopping as: Ni 0.078 eV, Cu 0.089 eV, Pd 0.111 eV, Ag 0.075 eV, and Pt 0.113 eV. These values are not in good agreement with previous estimates. Experiments, but also the many calculations, leave the activation energies for diffusion on gold in some doubt, and more definitive studies are clearly in order.

5.27

Lead: Pb(111) Regrettably there is hardly any information in the literature about diffusion on lead surfaces. However, Li et al. [372] did carry out calculations for surface self-diffusion on Pb(111) using EAM potentials and arrived at a barrier height of 0.045 eV.

5.28

Bismuth: Bi(111) Recently homoepitaxial growth of submonolayers of Bi on a Bi(111) surface was investigated by Jnawali et al. [373] using SPA-LEED and STM at temperatures in the range 80–200 K. Based on nucleation theory for a coverage of 0.5 ML, they came out with a value of 0.135 eV for the diffusion barrier on the flat bismuth terrace. There are no other data available to compare this value.

Jump

Al/Al(100)

Jump [100] Ex

Mechanism

System

ED (eV)

Tables for 2D diffusion

Table 5.7 Fcc(100).

5.29

Do (cm2/sec)

Experiment

0.8 0.65 0.4 0.46 0.6 0.45–0.58 0.692 0.74 0.30 0.51 0.57 0.20 0.53 0.52 0.5 0.60 0.46 0.67 0.81 0.20 0.25 0.69 2.28 0.1–0.24 0.271 0.35 0.27 1.11 0.56

ED (eV)

9.4 × 102

9 × 10−2

4 × 10−2

6.1 × 10−3

2.4 × 10−3

2 × 10−3 1.5 × 10−2 1 × 10−2

Do (cm2/sec)

Theory ν0 (sec−1) LDA LDA EAM AFW EAM VC L-J EAM EMT DFT LDA EMT*** DFT GGA DFT RGL EAG LDA VASPN GGA VASPN L-J VASP VASP DFT LDA LDA EAM VC EAM AFW L-J EAM EMT DFT DFT LDA RGL EAG

Technique

[1] [2] [4] [4] [4] [7] [9] [11] [12] [13] [19] [20] [18] [21] [21] [26] [27] [28] [11] [2] [4] [4] [4] [7] [9] [10] [11] [20] [18]

Ref.

Ni/Ni(100)

System

Table 5.7 (cont.)

Jump

L-Ex A-Ex

Mechanism

0.63£ 0.43 0.60 ± 0.02

ED (eV)

Do (cm2/sec)

Experiment

0.3 0.63 0.68 0.80 0.61 0.63 0.64 0.70 0.45 0.93 0.53 0.298 0.67 0.60 0.74 0.558 0.67 0.632 0.63 0.68 0.631 0.72Ã 0.63 0.20 0.61

0.23 0.14 0.17 0.13 0.39–0.55 0.85

ED (eV)

3.4 × 10−4 1.3 × 10−3

3.5 × 10−3 3.7 × 10−3 3.5 × 10−3 6.4 × 10−3 2.87 × 10−3 1.06 × 10−3

1.6 × 10−3 5.4 × 10−3 6.9 × 10−3 3.2 × 10−3

Do (cm2/sec)

Theory

3.71 × 1012

~1012 † 1012

ν0 (sec−1) Ref. [21] [21] [27] [28] [7] [18] [54] [74] [73] [47] [4] [4] [4] [76] [76] [56] [56] [56] [56] [60] [60] [29] [29] [7] [61] [78] [79] [80] [80] [81] [13] [82] [49] [84,85]

Technique LDA VASPN GGA VASPN VASP DFT EAM EAG FIM STM WF Morse EAM AFW EAM VC L-J CEM CEM¥ EAM FDBS EAM VCS EAM OJS EAM ATVFS SCM Morse CEM CEM¥ EAM EMT EAM EAM EAM FDB EAM VC EMT DFT EAM Coh. EAM

Ex Jump

Ex Jump

Rh/Ni(100)

Pd/Ni(100)

Ag/Ni(100)

Cu/Cu(100)

Au/Ni(100)

Jump

Ex Jump

0.40 0.48

0.31 ± 0.03 0.69

Ex Jump

Ex

0.35 ± 0.02

L-Ex Jump

Cu/Ni(100)

In/Ni(100) Pt/Ni(100)

0.59 ± 0.03

Ex

1 × 10−3

1.4 × 10−4 2.3 × 10−3

†

1 × 10−5.2 ± 0.6

0.55 0.706 0.48 0.489

0.62 0.45 0.407 1.23 0.67 0.65 0.85 0.55 0.64 0.492 1.10 0.42 0.44 0.347 1.47

0.93 1.15 0.47 1.0 0.85 1.29 0.844 0.78 1.304 1.42

0.82 0.75 0.376 4 × 10−2

2.5 × 10−3

2.9 × 1012

3.7 × 1012

4.45 × 1012

11.9 × 1012 10 × 1012 7.8 × 1012

3.76 × 1012 4.5 × 1012 8.28 × 1012 9.37 × 1012 4.2 × 1012 9.77 × 1012

4 × 1011 ± 0.3 5.2 × 1012 4.15 × 1012

7.28 × 1012

DFT L-J RGL FIM EAM AFW EAM VC CEM CEM¥ EAM EAM EMT DFT RGL EAM STM CEM CEM¥ RGL CEM¥ CEM¥ CEM¥ CEM¥ CEM¥ CEM¥ RGL CEM¥ CEM¥ CEM¥ RGL CEM¥ PACS FIM CEM¥ RGL CEM¥ RGL He-Scat He-Scat

[86] [26] [70] [55] [4] [4] [29] [29] [7] [78] [81] [86] [70] [7] [87] [76] [89] [70] [89] [76] [89] [89] [76] [89] [70] [89] [76] [89] [70] [89] [88] [72] [76] [70] [76] [70] [90] [91]

System

Table 5.7 (cont.)

Jump

Mechanism

0.34 ± 0.07 0.28 ± 0.06 0.39 ± 0.06 0.36 ± 0.03 0.36

ED (eV) ~10−5

Do (cm2/sec)

Experiment

0.248 0.27 0.38 0.53 0.43 0.66 0.49 0.398 0.51 0.53 0.46 0.243 0.249 0.425 0.45 0.69 0.48 0.399 0.503 0.52 ± 0.04 0.43 ± 0.02 0.52 ± 0.05 0.49 ± 0.01 0.51 0.53 0.93 0.49 0.51 ± 0.02

ED (eV)

3.2 × 10 3.3 × 10−4

−4

1.6(±0.3) × 10−3 3.4 × 10−3 ± 0.2

8.62 × 10−3 7.80 × 10−3

4.7 × 10−4

1.2 × 10−3 5.2 × 10−3

Do (cm2/sec)

Theory

1 × 1013

12 18:2þ4:03
3:29 × 10

20(× e±0.2)×1012

5.5 × 1012

†

ν0 (sec−1) STM He-Scat LEIS HR LEED PACS Morse Morse EAM AFW EAM VC EMT CEM¥ EAM EMT CEM¥ EAM VC CEM SCM Morse EMT EAM DFT EAM EMT EAM CEM¥ RGL DFT GGA EAMM EAM FDB EAM VC LDA EAM Thermo

Technique [93] [92] [94] [95] [96] [97] [47] [4] [4] [98] [76] [100] [99] [29] [77] [29] [102] [102] [61] [103] [104] [105] [107] [79] [108] [111] [112] [112] [80,114] [80,114] [11] [116] [113]

Ref.

Jump [100] β Ex

0.01 0.44 0.50 0.488 0.44 0.67 0.55 0.59 0.48 0.49 0.64 0.505 0.67 0.477 0.505 1.25 0.71 ± 0.05 0.72 0.79 ~0.22 ~0.20 0.77 0.79 0.18 0.97 0.80 0.996 0.7 ± 0.04 0.96 ± 0.1 0.70 ± 0.04 0.69 0.74 ± 0.02 0.70 ± 0.04 0.85 0.75 0.94 0.81 0.69 42 × 10−3 ± 0.3

8.5 × 10−4

4 × 10−2

38 × 10−3 ± 0.5

5.3 × 10−3

7.29 × 10−4

2.3 × 10−3

(2.5 ± 0.1) × 10−3

14 6:65þ5:47
3:00 × 10

437(× e±0.7) × 1012

Coh. RGL EAM EAM RGL DFT LDA DFT GGA L-J MS EAM DFT VASP EAM LDA RGL EAM LDA RGL EAM AFW EAM VC EMT EMT CEM¥ EAM VC CEM DFT EAM EMT RGL DFT GGA EAMM EAM Thermo RGLM RGLS EAM DFT LDA DFT GGA EAM

[49] [119] [121] [374] [67] [21] [21] [26] [122] [123] [28] [125] [127] [70] [126] [11] [111] [4] [4] [98] [99] [29] [77] [29] [104] [105] [81] [111] [112] [112] [116] [113] [119] [119] [121] [21] [21] [122]

Jump Jump

Ex Jump

Ex Jump

Ex Jump

Rh/Cu(100)

Pd/Cu(100)

Ag/Cu(100)

Ex

Jump

Jump

Ex [110] D-Ex T-Ex Q-Ex

Mechanism

Co/Cu(100)NMG Ni/Cu(100)

Co/Cu(100)

Na/Cu(100)

System

Table 5.7 (cont.)

0.058 ± 0.007

0.051 ± 0.006 0.0525 ± 0.0009 0.075

ED (eV)

2

(6.3 ± 0.6) × 10−3

Do (cm /sec)

Experiment

45 × 10−3 ± 0.6 104 × 10−3 ± 1.0 86 × 10−3 ± 0.9

1.02 0.94 1.02 0.708 2.81 0.70 ± 0.05 0.82 ± 0.08 0.75 ± 0.01

0.065 ± 0.006 0.61 0.66 1.00 0.86 0.92 0.64 0.64 0.439 0.57 0.71 0.72 0.42 0.58 0.73 0.566 0.59 0.36 0.48

0.077

Do (cm2/sec)

ED (eV)

Theory

15.58 × 1012 6.9 × 1012 9.07 × 1012

9.00 × 1012 4.6 × 1012 12.43 × 1012 10.61 × 1012 4.6 × 1012 17.57 × 1012

5.2 × 1012 9.44 × 1012

(0.53 ± 0.2)×1012

ν0 (sec−1) DFT VASP LDA RGL LDA RGL RGL RGL He-Scat He-Scat He-Scat LMD 3 He-SE LMD VASP RGL VASP RGL VASP CEM¥ CEM¥ RGL CEM¥ CEM¥ CEM¥ CEM¥ CEM¥ CEM¥ RGL CEM¥ CEM¥ CEM¥

Technique [123] [28] [127] [70] [11] [111] [111] [111] [133] [132] [134] [135] [136] [132] [123] [139]. [123] [139]. [123] [76] [89] [70] [89] [76] [89] [89] [76] [89] [70] [89] [76] [89]

Ref.

Jump

Jump

Cs/Cu(100)

Pt/Cu(100)

Au/Cu(100)

Ni/Rh(100)

Pb/Cu(100) Rh/Rh(100)

Ex Jump

In/Cu(100)

Jump

Jumpn Jumpm Ex

Jump Jump

Ex

Ex Jump

0.87 ± 0.07 0.83 ± 0.05 0.84

0.31 ± 0.02

0.24 ± 0.03 0.28 ± 0.05 0.20 ± 0.03

1 × 10−3 † 10−3 † 10−3 0.7 0.87 ± 0.16 1.094 0.388 ± 0.002 0.80 0.832 0.799 0.44 0.89 1.02 0.99 1.03 ± 0.01 0.92 ± 0.12 > 1.11 1.20 0.76 0.75

0.24 ± 0.04 0.61 0.874 0.56 0.554 0.64 0.58 ± 0.02 0.57 0.68 ± 0.1

0.32 1.01

0.393 0.91

6.1(×1.5±1) × 10−3 3.7(×1.78±1)×10−3

3.0 × 10−3

5.6(×2±1) x10−3 3 × 10−3 1.36 × 10−3 2.13 × 10−3

4.06 × (e±0.85) × 10−3

5.5 × 1012 5.82 × 1012

4.4 × 1012

3.1 × 1012

4.1 × 1012

7.5 × 1012

RGL CEM¥ LEIS PACS PACS EAM EAM 3 He-SE LMD CEM¥ RGL CEM¥ RGL RGLS RGL RGLS RBS FIM FIM FIM Morse L-J MCTST SCM TST CEM¥ SCM Morse Coh. DFT L-J DFT MCTST L-J MCTST L-J DFT DFT CEM¥ CEM¥

[70] [89] [128] [88] [129] [130] [130] [137] [137] [76] [70] [76] [70] [138] [138] [138] [131] [167] [169] [170] [47] [171] [175] [176] [76] [60,102] [60,102] [49] [179] [26] [28] [173] [174] [179] [28] [76] [89]

Ex Jump

Ex Jump

Ex Jump

Pd/Rh(100)

Ag/Rh(100)

Pt/Rh(100)

Ex

Jump

Jump

Ex Jump

Cu/Rh(100)

Au/Rh(100) Pd/Pd(100)

Mechanism

System

Table 5.7 (cont.)

0.59–0.63 0.51–0.62 ~0.65

0.92 ± 0.13

ED (eV)

2.0 × 10−3 ± 1.9

Do (cm2/sec)

Experiment

0.55 0.71 0.74 0.61 0.64 0.75 0.503 0.71 0.71 0.15 0.60 0.62 ± 0.04 0.87 0.621 0.7 0.61 0.59

0.66 0.55

1.21 0.74 0.53 1.44 0.66 0.73 1.46 0.50 0.51 1.91

ED (eV)

3 × 10−2

1.1 × 10−3

1.2 × 10−3 6.0 × 10−3

Do (cm2/sec)

Theory

5 × 1012

5.51 × 1013

4.2 × 1012

†

3.7 × 1012 2.7 × 1012 † 5 × 1012

6.5 × 10 5.4 × 1012 2.46 × 1012 3.62 × 1012 4.1 × 1012 4.07 × 1012 8.64 × 1012 4.9 × 1012 3.31 × 1012 4.14 × 1012

12

ν0 (sec−1) CEM CEM¥ CEM¥ CEM¥ CEM¥ CEM¥ CEM¥ CEM¥ CEM¥ CEM¥ FIM CEM¥ CEM¥ LEED RD Rean Morse EAM AFW EAM VC CEM¥ CEM CEM¥ EMT EAM DFT Coh. L-J EAM VASP RGL EAM EAM AFW EAM VC

¥

Technique [89] [76] [89] [89] [76] [89] [89] [76] [89] [89] [182] [76] [76] [185] [186] [186] [47] [4] [4] [76] [77] [77] [61] [82] [179] [49] [26] [187] [28] [70] [78] [4] [4]

Ref.

Ex Jump

Cu/Pd(100)

Rh/Pd(100)

Ag/Ag(100)

Au/Pd(100)

In/Pd(100) Pt/Pd(100)

Jump

Jump

Jump

Ex

Jump

Ex

Ex Jump

Ag/Pd(100)

Jump

Ni/Pd(100)

0.45 0.40 ± 0.05 0.330 ± 0.005 0.38 0.40 ± 0.04 0.40 0.35 ± 0.03

0.45 ± 0.04

0.37 ± 0.03

0.43 0.64 0.985 0.54 0.59

0.66 0.42 0.50 0.60 0.459 1.34 0.75

0.70 0.96 0.82 0.85 0.725 0.51 0.71 0.70 0.566 0.78 0.72 0.52 0.558 0.98 0.75 0.75 0.70

1 × 1012 1012 † 1013 † 3 × 1013 5 × 1012 † 1012 ± 1 †

†

2.8 × 1012

1013 4.0 × 1012

†

6.38 × 1012

8 × 1012 4.96 × 1012

3.98 × 1012 4.3 × 1012 7.85 × 1012 1.376 × 1013 8 × 109

9.74 × 1012 5.3 × 1012 2.63 × 1012

3.41 × 1013 5.2 × 1012 7.01 × 1012

CEM CEM¥ DFT VASP RGL EAM CEM¥ CEM¥ RGL CEM¥ CEM¥ CEM¥ RGL CEM¥ CEM¥ CEM¥ CEM¥ He-Scat EAM CEM¥ CEM¥ EAM RGL CEM¥ EAM PACS LEED CEM¥ RGL CEM¥ RGL SEM LEIS STM STM STM STM PACS

[29] [29] [179] [28] [70] [78] [76] [89] [70] [89] [76] [89] [70] [89] [76] [89] [89] [189] [189] [76] [89] [191] [70] [89] [191] [88] [188] [76] [70] [76] [70] [200] [223] [224] [225] [227] [229] [228]

System

Table 5.7 (cont.)

Ex

Jumpz

Mechanism

0.45 ± 0.03 0.48 ± 0.03

0.37 ± 0.03

ED (eV)

2

Do (cm /sec)

Experiment

0.75

0.35 0.49 0.48 0.48 0.24 0.45 0.41 0.479 0.365 0.50 ± 0.03 0.48 0.52 0.45 0.478 0.367 0.48 0.48 0.43 ± 0.04 0.46 0.48 0.26 ± 0.04 0.06 0.43 0.52 0.60 0.62 0.467 0.49 0.50

ED (eV)

3.4 × 10−4 5.5 × 10−4

2.2 × 10−3

8.1 × 10−4 2.3 × 10−3 (1.4 ± 0.6) × 10−3

1.2 × 10−3 3.9 × 10−3

Do (cm2/sec)

Theory

1.53 × 1013

1.0 × 1012

ν0 (sec−1) PACS Morse EAM EAM AFW EAM VC CEM¥ CEM¥ CEM EAM Univ.3 EMT Muffin tin EAMM DFT LDA DFT GGA EAM EMT EAM FDB EAM VC RGLM RGLS EAM SEAM Coh. RGLN L-J DFT LDA RGL EAM FDB EAM VC PACS PACS EAM AFW

Technique [96] [47] [230] [4] [4] [76] [77] [77] [207] [61] [208] [78] [231] [231] [79] [81] [233] [233] [212] [212] [82] [119] [49] [214] [26] [28] [127] [70] [80] [80] [228] [96] [4]

Ref.

Jump

Ex Jump

Ex Jump Jump

Jump

Ex

Rh/Ag(100)

In/Ag(100) Pt/Ag(100)

Au/Ag(100)

Ir/Ir(100)

Jump

Ex

Ex

D-Ex Jump

Cu/Ag(100)

Pd/Ag(100)

Ni/Ag(100)

0.93 ± 0.04

0.31 ± 0.03

0.43 ± 0.05

1.4×10−2±1

0.59 1.189 0.54 0.662

0.53 0.38 0.58 0.76 0.674 0.73 0.55 0.569 0.52 0.74 0.73 0.26

0.60 0.70 0.41 0.78 0.93 0.73 0.614 0.59 ± 0.02 0.39 0.76 1.00 0.624 0.74 ± 0.06 0.71 0.70 0.61 0.529 0.33 0.19 (300 ± 200) × 10−3

(40 ± 10) × 10−3

2 × 10−2

5 × 1012 ± 0.5

3.2 × 1012

14.44 × 1012 4.5 × 1012 9.69 × 1012 3.43 × 1012 † 1012 3.7 × 1012

5.6 × 1012 34.02 × 1012

4.01 × 1012 5.0 × 1012 16.08 × 1012

†

7.29 × 1012

5.0 × 1012 18.24 × 1012

3.92 × 1014

EAM VC CEM¥ CEM EAMM DFT LDA DFT GGA EMT RGLM SEAMS DFT LDA RGL RGLM CEM¥ CEM¥ EAM RGL CEM¥ EAM Photo EAM CEM¥ CEM¥ CEM¥ RGL CEM¥ CEM¥ RGL CEM¥ CEM¥ CEM¥ CEM¥ PACS CEM¥ RGL CEM¥ RGL FIM

[4] [29] [77] [78] [231] [231] [81] [212] [119] [28] [127] [70] [212] [76] [89] [239] [70] [89] [239] [237] [237] [89] [76] [89] [70] [76] [89] [70] [89] [76] [89] [89] [235] [76] [70] [76] [70] [246]

Rh/Ir(100) Pt/Pt(100)

System

Table 5.7 (cont.)

Jump

Jump Ex

Jump

Mechanism

2.09 × 10−3 ± 1.5 1.3×10−3±1

6.26×10−2±1 7.30 × 10−3 ± 0.5

0.84 ± 0.05 0.74 ± 0.02

0.80 ± 0.08 0.47 ± 0.01

Do (cm /sec)

2

ED (eV)

Experiment

0.31 0.64 3.97 0.611 ± 0.004 0.54 0.38 0.865 0.98 ± 0.04 0.99 0.27 ± 0.05 0.31 0.388 0.8 0.44 1.25 1.05 0.82

0.79 0.77 0.51 0.59 ~0.97 1.58 1.57 1.39 ± 0.04 1.13 0.40 1.23 1.17

ED (eV)

8 × 10−4 5 × 10−3 4.9 × 10−3

1 × 10−2 4 × 10−2 3.5(×0.2±1)×10−2

2.6 × 10−3

Do (cm2/sec)

Theory ν0 (sec−1) FIM FIM EAM RGL DFT DFT Morse EAM RGL Muffin tin DFT Coh. L-J DFT FIM FIM EAM AFW EAM VC L-J SC EAM DFT OJ EAM EAMS EAM VASP RGL Morse EAM AFW EAM VC L-J Morse

Technique [315] [307] [309] [309] [179] [28] [47] [309] [309] [208] [179] [49] [26] [28] [307] [345] [4] [4] [4] [176] [323] [179] [350] [119] [119] [354] [28] [70] [47] [4] [4] [4] [4]

Ref.

A-Ex Jump Jump

Jump

Jump Jump

Jump

Jump

Jump

Pt/ Pt(100) Hex Ni/Pt(100)

Cu/Pt(100)

Rh/Pt(100) Pd/Pt(100)

Ag/Pt(100)

Au/Pt(100)

Au/Au(100)

0.70 ± 0.01

0.43 ± 0.03

†

1 × 10

−3

0.49 0.627 0.41 0.508 0.58 0.507 0.6 0.64 0.84 0.67 0.49 0.43 0.62 ± 0.04 0.49 0.50 0.83 0.58 0.70 −0.03 0.53 0.64

0.57 0.686 0.61 0.615 0.57

0.72 0.689 1.04 0.07 1.84 1.02 0.36 ± 0.05 1.2 0.875 0.96

1.5 × 10−3

1.56 × 10−2

8 × 10−4 3.5 × 10−3

2.1 × 10−3

2.11 × 1013

2.2 × 1012

2.1 × 1012

9.3 × 1012

3.2 × 1012

3.3 × 1012

4.5 × 1012

4.5 × 1012

2.5 × 1012

CEM¥ EMT DFT Coh. EAM L-J EAM VASP RGL EAM STM CEM¥ RGL CEM¥ RGL CEM¥ FIM CEM¥ RGL CEM¥ RGL CEM¥ RGL Morse EAM AFW EAM VC CEM¥ EMT EAMM Muffin-tin EAMM EAM TST DFT LDA DFT GGA EAM Coh. RGL L-J [76] [61] [179] [49] [119] [26] [354] [28] [70] [119] [348] [76] [70] [76] [70] [76] [355] [76] [70] [76] [70] [76] [70] [47] [4] [4] [76] [61] [78] [208] [359] [359] [361] [361] [82] [49] [214] [26]

Jump Jump

Jump

Jump

Rh/Au(100) Pd/Au(100)

Ag/Au(100)

Pt/Au(100)

 0.2 0.35 0.4

ED (eV)

2

5 × 10−8

Do (cm /sec)

Experiment

0.15–0.17 0.28 0.61 0.591 0.66 0.547 0.62 0.53 0.582 0.32 0.437 0.82 0.994

0.86 0.64 0.531 0.30 0.32 0.25 0.26 0.28 0.65 0.40 0.41 0.32 0.60 0.388

ED (eV)

5.8 × 10−3

1 × 10−2

Do (cm2/sec)

Theory

2.5 × 1012

5.8 × 1012

3.6 × 1012 3.7 × 1012

5.0 × 1012

4.6 × 1012

9.78 × 1012

ν0 (sec−1) DFT LDA RGL EAM AFW EAM VC EAMM EAMM EAMS DFT LDA DFT GGA RGL DFT LDA RGL STM Analysis AFM EMT EMT CEM¥ RGL CEM¥ RGL CEM¥ CEM¥ RGL CEM¥ RGL CEM¥ RGL

Technique [28] [127] [70] [4] [4] [78] [359] [359] [361] [361] [214] [28] [127] [70] [356] [357] [358] [360] [360] [76] [70] [76] [70] [76] [76] [70] [76] [70] [76] [70]

Ref.

Ã

*** on cluster; M molecular dynamics; S static; z 600 K; m Wide Temp.; n Low Temp.; NMG non-magnetic; † assummed value; £ assumed kT‘2 /2h; N NEB; paramagnetic; CEM¥ -- MD/MC CEM

Jump

x-Jump y-Jump

Ex

Mechanism

Cu/Au(100)

Ni/Au(100)

Au/Au(100) Hex

System

Table 5.7 (cont.)

Mechanism

Jump

Ex Jump

System

Al/Al(111)

Ni/Ni(111)

Table 5.8 Fcc(111).

Do (cm2/sec)

0.33£ 0.22 ± 0.02

0.07 ± 0.01 1 × 10−5.2 ± 0.7

0.042 ± 0.004 2 × 10−9 ± 0.25 0.06 0.04

ED (eV)

Experiment

~0.01 0.056 0.063 0.15 0.06 0.07 0.05 0.05 0.086 0.016 0.068 0.036 0.16 ± 0.02

0.63

0.02 0.04 0.03 0.042 0.067 0.13 0.04

0.074 0.054 0.11 0.04 0.04

ED (eV)

8.74 × 10−4 5.04 × 10−4

5 × 10−4 6.2 × 10−4 9.0 × 10−4

1.4 × 10−3 3.2 × 10−4

9 × 10−4 1.6 × 10−3 1.3 × 10−3

Do (cm2/sec)

Theory

5 × 1011 ± 0.5

4 × 1012

1 × 1011 5 × 1011

ν0 (sec−1) EAM AFW EAM VC L-J DFT DFT STM STM STM EMT*** DFT EAM DFT RGL L-J VASP STM VASP FIM FIM Morse EAM AFW EAM VC L-J EAM FBD EAM VC EAM OJ EAM ATVF SC Morse EMT CEM DFT LDA

Technique

[4] [4] [4] [10] [31] [32] [33] [34] [12] [35] [35] [19] [20] [26] [27] [38] [27] [54] [55] [47] [4] [4] [4] [56] [56] [56] [56] [60] [60] [61] [62] [63]

Ref.

Jump Jump Jump Jump

Jump Jump

Au/Ni(111) Cu/Cu(111)

Ex

Mechanism

Cu/Ni(111) Pd/Ni(111) Ag/Ni(111) Pt/Ni(111)

System

Table 5.8 (cont.)

0.037 ± 0.005 0.040 ± 0.001

0.03þ0:01
0:005 0.064

0.2–0.3

ED (eV)

Do (cm2/sec)

Experiment

0.05 0.023–0.032 0.026 0.044 ~0.12 0.059 0.064 0.14 0.053 0.028 0.039

0.057

0.018 0.038

0.11 ± 0.02 0.06 0.05 0.063 0.14 0.049 0.061 >2.0 2.15 1.633 0.050 0.045 0.038

ED (eV)

6.4 × 10−4 5.24 × 10−4

3 × 10−4 4.6 × 10−4

(1.8 ± 0.1)×10−4 1.7 × 10−4

Do (cm2/sec)

Theory

5 × 1013 ± 1 1 × 1012 ± 0.5 1 × 1012

ν0 (sec−1) DFT GGA EAM EAM EAM L-J 2nd Moment RGL 2nd Moment EAM RGL RGL RGL RGL FIM RGL RGL He-Scat SPA-LEED EMT STM* STM* DACAPO Morse EAM AFW EAM VC EMT SC Morse Scaling EMT EAM CEM

Technique

[63] [35] [64] [67] [26] [68] [70] [68] [64] [70] [70] [70] [70] [72] [70] [70] [140] [141] [142] [143] [144] [145] [97] [4] [4] [98] [102] [102] [15] [61] [105] [62]

Ref.

Na/Cu(111)

Li/Cu(111)

αfh αhf αfh

αhf

Exr αfh

Jump Jumps Jump [110]t Jump [110]s Jump ? [110]t Jump ? [110]s Ex

t

0.16 ± 0.02 0.12 ± 0.02 0.029 0.05 0.05 0.05 0.042 0.09 0.11 0.036 ± 0.002 0.030 ± 0.001 0.043 0.026 0.01 0.043 0.06 0.041 ± 0.002 0.087 ± 0.008 0.043 ± 0.001 0.082 ± 0.007 0.040 ± 0.001 0.10 ± 0.01 1.12 0.5 1.455 1.42 1.352 1.19 0.13 0.03 0.041 0.04 0.03 0.11 0.027 0.20 0.054 0.30 2.06 × 10−4 6.49 × 10−5 4.54 × 10−5 5.32 × 10−5

1.84 × 10−4

3.01 × 10−4 (2 ± 0.1)×10−4 (7 ± 1)×10−4

1.4 × 10−4 1.7 × 10−4 1.56 × 10−4

(1.2 ± 0.1) × 10−4

DFT LDA DFT GGA EAM Coh. DF EAM RGL Emp. L-J EAM Msd EAM 2nd Moment EMT EAM RGL EAM RGL RGL RGL RGL RGL RGL EAM EMT 2nd Moment EAM RGL RGL EAM EAM EAM EAM EAM EAM EAM LDA LDA LDA

[63] [63] [116] [49] [149] [35] [67] [150] [26] [155] [155] [68] [124] [117] [70] [156] [146] [146] [146] [146] [146] [146] [105] [98] [68] [117] [70] [146] [148] [126] [154] [153] [122] [148] [126] [158] [158] [158]

Pt/Rh(111)

Rh/Rh(111)

Pb/Cu(111)

Pt/Cu(111) Au/Cu(111)

Ni/Cu(111) Pd/Cu(111) Ag/Cu(111) In/Cu(111)

K/Cu(111)

System

Table 5.8 (cont.)

Jump Jumpm Jump

n

Jumpp Jump Ex Jump

αhf αfh αhf Jump Jump Jump Jump Jump Ex Jump Jump

Mechanism

0.22

0.16 ± 0.02 0.18 ± 0.06

ED (eV)

†

1 × 10−3

2 × 10−4

Do (cm2/sec)

Experiment

0.05 0.27 ± 0.026 0.29 ± 0.03 0.035 0.092 ± 0.033 0.118 ± 0.043 0.234 0.106 0.089 0.27 ± 0.01 0.26 0.15 0.21 0.224 ± 0.003 0.20 ± 0.01

0.13 0.20 0.05 0.045 0.041 0.034 0.02 0.08 1.26 0.009 0.061 0.039 0.08 ± 0.01 0.03 1.17

ED (eV)

1.9 × 10−4 5.4(×1.37±1) × 10−4 2.91(×1.16±1)×10−4

7.10 × (e±0.25) × 10−4 7.1(×1.28±1) × 10−4 6.2 × 10−4 (4.6 ± 2.3) × 10−4 (7.1 ± 4.5) × 10−4 4.5 × 10−4 6.09 × 10−4 6.40 × 10−4 (1.1 ± 0.2) × 10−5

3 × 10−4

5.21 × 10 2.02 × 10−5 1.83 × 10−5

−5

Do (cm2/sec)

Theory ν0 (sec−1) LDA LDA LDA RGL RGL RGL EAM EAM EAM RGL RGLS RGL RGLM VASPN GGA VASPN GGA FIM STM Morse L-J MD CEM CEMM MorseM L-J SC Morse L-JM Coh. RGL L-J L-J MC MC TST FIM

Technique

[158] [158] [158] [70] [70] [70] [130] [157] [130] [70] [138] [70] [138] [159] [159] [167] [168] [47] [171] [173] [76,177] [76,177] [76,177] [76,177] [60,102] [60,102] [108] [49] [180] [26] [173] [173] [182]

Ref.

Jump

Jumpl Jumpg Ex Jump

Jump Jump Jump Jump Jump Jump Jump

Pd/Pd(111)

K/Pd(111)

Ni/Pd(111) Cu/Pd(111) Ag/Pd(111) 111 In/Pd(111) Pt/Pd(111) Au/Pd(111) Ag/Ag(111)

0.1

0.15 ± 0.10 0.12 0.18 0.051 ± 0.024 0.097 ± 0.010

0.58

0.23 ± 0.03

820K; j 140–300K; k 300–800K; l 150–500K; m Wide Temp.; n Low Temp.; M molecular dynamics; S static; ST strained surface; p T100K; s 400–900K; t 80–300K; £ assumed kT‘2 /2h; † assumed value; § two layers relaxed; *** on cluster; * direct measurement; N NEB; CEM¥ -- MD/MC CEM

Ni/Au(111) Cu/Au(111) Pd/Au(111) Ag/Au(111) Pt/Au(111) Pb/Pb(111) Bi/Bi(111)

Co/Au(111)pffiffiffi (22  3)

Co/Au(111)

Au/Au(111)pffiffiffi ST ð23  3Þ Al/Au(111)pffiffiffi ð22  3)

System

Table 5.8 (cont.)

N

Jump Jump Long α Short α Ex Short α Long α Ex Jump Ex

Jump Jump Jump

ExND

Jump [110] α ExD

1.01 ± 0.02 0.53-0.54 0.13 1.3

0.43 ± 0.04

2

† 2 × 10−3 2.7 × 10−4 – 3.3 × 10−4

8.7 × 10−2

(9.5 ± 0.9) × 10−4 7.2 × 10−4

Do (cm /sec)

Experiment

1.23 1.75 1.06 1.16 2.02 1.2 1.3 2.3 1.6 2.1

0.6 0.46 0.92 0.84 0.4o 0.5o 0.7o 0.66 0.60 0.99 0.97 1.4 1.0 1.63 0.49

ED (eV)

306 × 10−3

113 × 10−3

206 × 10−3

Do (cm2/sec)

Theory

9 × 1011 ± 0.7

ν0 (sec−1) He-Scat. STM He-Scat RHEED STM Morse Scaling EAMM EAMN VASP VASP VASP EAMM EAMN EAMM EAMN Morse Morse Morse Ackland SIM VASP VASP VASP VASP VASP VASP VASP VASP VASP VASP FIM FEM FEM FEM

Technique

NEB; o on 1ML Fe on W(100); R on reconstructed surface; MG magnetic; NMG non-magnetic; ND non-diagonal; D diagonal; † assumed value; M molecular dynamics.

Ni/W(100) Hf/W(100) Au/W(100) Pb/W(100)

Fe/W(100)

Fe/W(100) R

K/W(100) Mn/W(100)MG Mn/W(100)NMG Mn/W(100)R

Mo/Mo(100) Ta/Ta(100) W/W(100)

0.063 ± 0.015 0.45 ± 0.08 0.495 ± 0.050 0.37

Jump

K/K Fe/Fe(100)

Jump

ED (eV)

Mechanism

System

Table 5.9 Bcc(100).

[39] [40,41] [42] [43] [47] [15] [52] [52] [50] [50] [50] [52] [52] [52] [52] [47] [47] [47] [294] [290] [263] [263] [263] [263] [263] [50,51] [50,51] [50,51] [50,51] [50,51] [291] [292] [288] [293]

Ref.

Jump

W/Re(0001)

assumed value.

0.48 ± 0.01

αhf Jump Jump Jump Jump Jump Jump

Li/W(111) Ti/W(111) Ni/W(111) Pd/W(111) Dy/W(111) Re/Re(0001)

†

1.9 1.3 0.53 1.77 0.873 ± 0.081 1.02 ± 0.06 1.25 ± 0.09 0.48 ± 0.02

Jump Jump Jump Jump

Mo/Mo(111) Ag/Ru(0001) Ta/Ta(111) W/W(111)

~1.8 0.46

Jump

Fe/Fe(111)

ED (eV)

Mechanism

System

Table 5.10 Bcc(111) and Hcp(0001).

2.17(×2.7±1) × 10−3

1.30 × 10−2 3 × 10−2 1 × 10−6 † 1 × 10−3 103 6.13(×2.6±1) × 10−6

10−3

†

5.8 × 10−8

Do (cm2/sec)

Experiment

1.01

2.92

1.4 0.18 1.9 0.057 2.2

ED (eV)

Do (cm2/sec)

Theory ν0 (sec−1) Morse Scaled Morse EAM Morse FIM FEM Morse FIM FIM FEM FEM FIM FIM FEM FIM Coh. FIM

Technique

[47] [15] [47] [166] [47] [296] [297] [47] [298] [296] [300] [301] [299] [272] [302] [16] [49] [16]

Ref.

0.49 ± 0.02 0.92 ± 0.05 0.90 ± 0.07 0.91 ± 0.03

0.92 ± 0.02 0.93 ± 0.01 0.94 ± 0.03 1.24 ± 0.13

Jump

Jump

Jump Jump Jump Jump Jump Jump

Jump Jump

x-eff Jump y-eff Jump α β

Mo/Mo(110)

Fe/Mo(110) Cu/Mo(110) Ga/Mo(110) Rh/Mo(110) In/Mo(110) Sn/Mo(110) Ba/Mo(110) Ta/Ta(110)

Pd/Ta(110) W/W(110)

0.1 ± 0.05 0.54 0.25 0.62 ± 0.06 0.30 0.50 0.7

Jump

Fe/Fe(110)

ED (eV)

Mechanism

System

Table 5.11 BCC(110).

−3

1.61(×1.6 )×10 5.21(×1.4±1)×10−3

±1

2 × 10−3 2.6 × 10−3 6.2(×13±1)×10−3 1.2(×9±1) × 10−2

†

8.7 × 10 2 × 10−5 1 × 10−3 3 × 10−5 4 × 10−5 20

−4

D0 (cm2/sec)

Experiment

0.44 0.54 0.46 0.52 0.70 ± 0.07 0.982 1.14 1.22 0.89

0.45 1.19

0.27 ~0.25 0.65 0.65 0.40 0.40 1.20

ED (eV)

6.92 × 10−3

1.3 × 10−3 1.1(×1.31±1)×10−3 3.6(× e±0.57) ×10−3 1.11 × 10−3

1.27 × 10−2

D0 (cm2/sec)

Theory

5.92(×2.5±1)×1012 8.06(×8.1±1) × 1015

8 × 1011 ± 1

ν0 (sec−1) F-S Morse Coh. DFT GGA EAMM Morse Coh. STM Spreading FIM FIM FIM FIM WF Morse Coh. FIM FIM Rean FIM Morse Morse Morse Morse L-J L-JM 4th Moment Coh. EAMM FIM FIM FIM FIM

Technique [44] [47] [49] [53] [46] [47] [49] [163] [161] [162] [160] [162] [162] [164] [47] [49] [240] [167,242] [245] [246] [248] [249] [47] [250] [174] [251] [252] [49] [253] [247] [247] [247] [247]

Ref.

Jump Jump Jump Jump Jump

Co/W(110) Ni/W(110)

Cu/W(110) Y/W(110) Zr/W(110) Nb/W(110) Mo/W(110)

0.49 0.49 ± 0.02

0.60 0.2-0.5

0.28 0.4

1.28 ± 0.13 1.37 ± 0.13

δx δy Ex

Jump Jump Jump Jump Jump Jump Jump α Ex [110] Ex [100] α Ex [100] Ex [110] Jump

ED (eV)

Mechanism

α Ex [110] Ex [001] Jump Jump

Fe/W(110)

Mn/W(110)NMG

Li/W(110) Na/W(110) K/W(110) Sc/W(110) Ti/W(110) V/W(110) Cr/W(110) Mn/W(110) Mn/W(110)MG

System

Table 5.11 (cont.)

†

†

10−3 2 × 10−3

2 × 10−8 10−3

D0 (cm2/sec)

Experiment

0.65 0.53 0.90 1.07 1.14 1.19

0.85 0.7 4.2 2.9 0.76

0.87 1.03 1.08 1.01 0.95 0.41 4.22 3.07 0.73 3.30 4.54

2.94 3.11 0.11

ED (eV)

1.78 × 10−3

D0 (cm2/sec)

Theory

3.73(×8.6±1)×1016 1.00(×8.4±1)×1018

ν0 (sec−1) FIM FIM 4th Moment EAMM WF FEM FEM Coh. Coh. Coh. Coh. Coh. VASP VASP VASP VASP VASP VASP LEED WF NRS Coh. VASP VASP VASP Coh. FIM FIM Coh. Coh. Coh. Coh. Coh. Coh.

Technique [247] [247] [252] [253] [257] [258] [262] [254] [254] [254] [254] [254] [263] [263] [263] [263] [263] [263] [264] [265] [254] [50] [51] [51] [254] [266] [267] [254] [254] [254] [254] [254] [254]

Ref.

Jump Jump

Jump

Jump Jump Jump

Jump

Jump Jump

x-eff Jump y-eff Jump α β δx δy Jump

Jump

Ag/W(110) Cs/W(110)

Ba/W(110)

Lu/W(110) Hf/W(110) Ta/W(110)

Re/W(110)

Os/W(110) Ir/W(110)

Pt/W(110)

Au/W(110)

0.94 ± 0.02 0.94 ± 0.02 0.94 ± 0.02 1.18 ± 0.04 1.25 ± 0.08 1.27 ± 0.09 0.67 ± 0.06 0.63

0.97 ± 0.02 0.95 ± 0.03

1.03 1.01 0.91

0.77 0.70

10−3

†

3.1(×10±1)×10−3 10−3

3.0(×1.8±1)×10−3 6.78(×1.8±1)×10−3

1.7(×2.3±1) × 10−2 8.48(×2.6±1)×10−3

†

3.0 × 10−2

4.4 × 10−2 10−3

†

0.20 0.23 3.2 × 10−8 1.3 × 10−4

1.4 × 10−3 ± 1.9 4.25(×1.7±1)×10−4

0.51 ± 0.03 0.509 ± 0.009

0.61 0.57 ± 0.02 0.41 0.078

2.5 × 10−3 1 × 10−3

†

0.72 ± 0.05 0.51

0.72 0.54

0.86

1.12 1.01

1.23

0.94 1.11

0.54

0.71

1.10 0.86

** multi atom investigation; NMG non-magnetic; MG magnetic; † assumed value; M molecular dynamics

Jump Jump Jump Jump

Tc/W(110) Ru/W(110) Rh/W(110) Pd/W(110)

5.45(×1.9±1)×1012 1.17(×4.2±1)×1015 2.70(×1.3±1)×1016 4.61(×1.8±1) × 1016

Coh. Coh. FIM FIM Coh. FIM FIM Coh. Desor Photo FEM WF Coh. Coh. FIM** FIM Coh. FIM** FIM FIM Coh. Coh. FIM FIM Coh. FIM FIM FIM FIM FIM FIM FIM FIM Coh. Coh.

[254] [254] [160] [266] [254] [272] [273] [254] [274,275] [276] [277] [278] [254] [254] [243] [279] [254] [280] [283] [279] [254] [254] [284] [285] [254] [286] [286] [286] [286] [286] [286] [287] [279] [254] [254]

402

Diffusion on two-dimensional surfaces

5.30

Predictions and comparisons In the previous sections we have focused on presenting diffusivities obtained on various faces for different elements. Much is still unknown, however, and the question therefore immediately arises how one can predict diffusion properties not yet ascertained in experiments. The answer is, of course, obvious: we have to rely on theory as a guide. This is an activity which may not have been stressed sufficiently in these presentations. It is clear, however, that advances in quantum physics as well as in computational capabilities have brought about great progress. Predictions about the energetics of diffusion abound in the literature, but how reliable are they? To get a feeling for this question, we will compare calculations with experimentally determined diffusivities of materials for which the measured results are considered highly reliable. To see how well the different theoretical approaches utilized in Chapters 4 and 5 work in giving us guidance about diffusivities, we will concentrate on self-diffusion for just a few systems: Cu(100) and Cu(111), Pt(111) as well as Ir(111), and finally W(110) and W(211). As is apparent from the previous sections, these surfaces have been thoroughly examined and there appears to be fairly uniform agreement about the experimental results. A number of different theoretical estimates are also available, so that some impression about the success of these efforts can be attained. The comparisons are made by presenting the results for the estimated activation energy, and sometimes also the prefactor for the diffusivity, which can then be contrasted with the experimental studies. First to be examined will be copper surfaces. Self-diffusion on Cu(100) has been studied experimentally by many groups, and although the results are not in perfect agreement, a reasonable value for the diffusion characteristics can be deduced. The record of theoretical estimates is clear from Table 5.7; they generally give much the same results independent of the theoretical approach. Furthermore, they agree more or less with the experimental values. What is worrisome, however, is the not too good agreement about the mechanism of movement, with later studies favoring hopping over exchange. On Cu(111) the experimental situation is much better, as recently two independent groups have arrived at essentially the same characteristics [143,144] directly measuring the movement of atoms. From Table 5.8 it appears that the number of attempts to calculate diffusion barriers is smaller than for Cu(100). For both (100) and (111) planes it is clear that theory and experiment are generally in reasonable agreement, independent of the approach used; this is especially true for some of the most recent estimates, but it must be remembered that in copper the d-shell is filled. That is why this agreement may not extend to other materials. It must also be emphasized that a 25% error in the diffusion barrier, not that uncommon in the calculated results, will have a very significant effect on the diffusivity. The next surface to be examined is Pt(111), in Table 5.8, also an fcc metal, again with a filled d-shell. The diffusivity has been studied independently by two different groups using different experimental approaches, and the two results agree nicely [305,321]. Kyuno did direct measurements of single atom movement by FIM and Bott looked at

5.30 Predictions and comparisons

403

kinetics of island formation modeled with Monte Carlo simulations. Platinum is the only material for which two different approaches agree so well. Now, however, it is clear from Table 5.8 that the semi-empirical potentials employed in calculational estimates generally give approximations which are really in discord with experiments, either much too high or too low. The ab initio calculations are too high, but the latest estimate is only 12% away from the experimental findings. The iridium (111) plane has been studied by several groups, but we adopt the work of Wang as representing reasonable experiments [304]. Only a few calculations have been made of iridium diffusivities, and it is clear from Table 5.8 that they vary a great deal and even the best differ from the experiments by 12%. Theoretical estimates have not come close enough to supplant experiments. Finally to be examined are the (110) and (211) surfaces of tungsten. For these, excellent measurements by Antczak are available [247,375], and are in reasonable agreement with other studies as is apparent in Table 4.4 and 5.11. However, there is only one attempt to do calculations using a modern approach, the work of Xu and Adams [252]. On the flat (110) plane the activation energy for diffusion calculated by Xu and Adams is ~ 24% higher than the value derived from experiments. For the W(211) plane, experiment and theory agree within the limit of error. Recently, using EAM potentials and molecular dynamics, a value of 0.89 eV was derived by Chen et al. [253] for the diffusion barrier in good agreement with experimental data. The value for movement on the W(110) surface are in quite good shape. The conclusions from these comparisons are clear. For copper, predictions agree well with experiments independent of the semi-empirical potential used. For the other systems, agreement can be achieved as well but to reach this goal careful choice of simulation settings is required and comparison with experiment is a key factor in this process. With fcc metals the RGL potential does reasonably well, but does not yield results of quantitative reliability. It seems that for quantitative information about diffusivities we are at the moment dependent upon experiments, which of late have not been intensively pursued. Calculated diffusivities are not usually reliable indicators of actual rates – they are, however, powerful for exploring novel mechanisms. There is another approach to gaining insight into what diffusivities could be like, which was suggested some time ago by Wang and Ehrlich [14] and has also been used by Kief and Egelhoff [15] as well as Feibelman [13], and that is to compare the measured activation energy with the enthalpy of vaporization of the diffusing element. The enthalpies are known for all the elements [376], and one might expect some correlations between diffusion characteristics for similar materials. That idea is tested in Table 5.12, in which are listed the surface examined, the activation energy for diffusion ED, the enthalpy of sublimation ΔHos , and finally the ratio Rs = ED /ΔHos ; it is hoped that the last quantity will give us some help in making a guess about the diffusivity, inasmuch as the prefactor for the diffusivity can be expected to be around 10−3 cm2/sec. In this listing we need to be particularly careful about fcc(100) and (110) surfaces; on these diffusion may occur either by an atom hopping or by exchange with an atom from the substrate, and these different mechanisms can be expected to give quite different ratios Rs. There is also some uncertainty in the measured results which can vitiate comparisons.

404

Diffusion on two-dimensional surfaces

Table 5.12 Comparison of activation energies and enthalpies of sublimation. Surface

Rh(111)

Adatom ED (eV) ΔHos (eV) Rs

Rh 0.16 5.754 0.0278

Surface Adatom ED (eV) ΔHos (eV) Rs

Ta(110) Pd W 0.49 0.92 3.908 8.776 0.125 0.105

Surface Adatom ED (eV) ΔHos (eV) Rs

W(110) Pd Rh 0.51 0.72 3.908 5.754 0.131 0.125

Surface Adatom ED (eV) ΔHos (eV) Rs

W 0.81 8.776 0.0923

Surface Adatom ED (eV) ΔHos (eV) Rs

Rh(100)

Pt 0.22 5.841 0.0377 W 0.92 8.776 0.105

Rh(311)

Pd(111)

Rh 0.87 5.754 0.151

Pt 0.92 5.841 0.158

Rh 0.54 5.754 0.0938

Pt 0.44 5.841 0.0753

Pd 0.350 3.908 0.0896

K 0.066 0.920 0.0717

Cs 0.57 0.790 0.722

Re 1.03 7.955 0.129

W(110) Ta 0.77 8.080 0.953

Ni 0.49 4.440 0.110

Ir 0.94 6.874 0.137

Pt 0.67 5.841 0.115

Ni 0.873 4.440 0.197

W(111) Li Ti 0.53 1.77 1.646 4.887 0.322 0.362

Pd 1.02 3.908 0.389

Re(0001) Re W 0.48 0.48 7.955 8.776 0.0603 0.0547

Rh 0.536 5.754 0.0932

Re 0.83 7.955 0.104

W(211) Mo Ir 0.71 0.67 6.80 6.874 0.104 0.0975

Ni 0.46 4.440 0.104

Ir 0.289 6.874 0.0420

Ir(111) Re 0.52 7.955 0.0654

W 0.51 8.776 0.0581

Ir(100) Ir Rh 0.74 0.80 6.874 5.754 0.108 0.139

Pd 0.32 3.908 0.0819

W(321) W 0.81 8.776 0.0923 Pt(111) Pt Ag 0.260 0.168 5.841 2.944 0.0445 0.571

Correlations are certainly evident between the ratios Rs = ED/ΔHos for neighboring materials in Table 5.12. For example, on Ta(110), the ratio Rs is 0.125 for diffusion of palladium; on W(110), a neighbor in the periodic table, this ratio is 0.131 for palladium. Furthermore, for a given substrate, this ratio is fairly constant from one adatom to another, with the exception of alkali metal adatoms. This approach must, however, be used cautiously, as is clear comparing Re(0001) with W(110), which for self-diffusion have Rs values 0.0603 compared to 0.105. More experiments are clearly in order to obtain usable estimates of diffusivities. The emphasis has so far been on diffusion barriers. Also necessary for predicting diffusivities are prefactors, and for these it is quite common in both experiment as well as theory to assume a standard prefactor for the diffusivity in the range 10−3cm2/sec or a standard frequency factor in the range 1012/sec. This assumption is probably appropriate if the atom is moving over the surface by hopping between nearest-neighbor places. However, it is still not clear how this value is influenced by the existence of long jumps, discussed in Chapter 3, or correlated motion on the surface. Such complicated mechanisms might change the values of the prefactor for the diffusivity, but experiments so far

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6

Diffusion in special environments

Single adatom diffusion on a variety of one- and two-dimensional surfaces has now been surveyed. However, events occurring at plane edges and other types of defects play an important role in affecting diffusion over the entire surface. There are also a huge number of indirect indications that impurities influence atom movement as well as other processes. The amount of direct information available for this field is not large, but we will examine it to gain at least some insight into such phenomena.

6.1

Near impurities Some effort has been made to uncover the way in which atomic diffusion is affected by impurities in the substrate. A start was made by Cowan and Tsong [1] in 1977, who explored the effect of rhenium atoms dissolved in tungsten on the surface diffusion of tungsten atoms on the (110) plane. This surface was prepared from W-3% Re alloy, and so had a number of rhenium substitutional atoms on the (110) plane. Tungsten atoms were then deposited on the surface, and the spatial distribution of places visited by atoms during their thermal movements was measured and compared with that for tungsten moving on the clean W(110), as shown in Fig. 6.1. Tungsten adatoms diffusing over a W-3% Re(110) surface were found to spend more time at a few sites on the surface, which the authors associated as most likely being next to interstitial rhenium atoms embedded in the surface. They estimated an increase in the binding energy of 0.090 ± 0.007 eV for a tungsten adatom in the nearest-neighbor position along the [001] direction from a rhenium atom embedded in the substrate. In the next-nearest-neighbor position, along the [110] direction, repulsion was equal to or larger than 0.080 eV. Cowan and Tsong also examined the kinetics of atom motion close to rhenium atoms, and found a diffusion barrier 0.041 ± 0.006 eV higher than on a clean part of the surface. The difficulty with this type of study is lack of exact knowledge of the location of the impurity atoms. This was remedied in 1994 by Kellogg [2], who embedded an iridium atom on a rhodium(100) surface by the exchange mechanism, as in Fig. 6.2. In this process, a rhodium adatom is produced on the surface and since the position of the embedded Ir atom is exactly known, details of the association of rhodium with the bound iridium, as well as changes in diffusion characteristics associated with the isolated impurity atom, can be studied. It must also be noted that iridium atoms on Rh(100) can be distinguished from

424

Diffusion in special environments

Fig. 6.1

Distribution of tungsten atoms over W(110) surface. (a) On clean (110) plane. (b) On (110) plane of W-3%Re after diffusion at 340 K. Concentration at specific sites is evident (after Cowan and Tsong [1]).

Fig. 6.2

FIM Ne images at 77 K show embedding of Ir atom on Rh(100). (a) – (b) Ir adatom exchanges with Rh substrate. (c)–(f) Field evaporation of outermost Rh plane reveals Ir atom embedded in lattice (after Kellogg [2]).

rhodium atoms as the voltage for their field evaporation is considerably lower than for rhodium; rhodium at a site adjacent to a substitutional iridium desorbs at an intermediate value. It turned out that a rhodium adatom was bound at four sites around iridium, as illustrated by the field ion microscope images in Fig. 6.3. From studies at temperatures 248 K to 265 K, Kellogg estimated a barrier for diffusion around the

6.1 Near impurities

Fig. 6.3

425

FIM images reveal the trapping of a Rh atom by an Ir atom substituted into Rh(100) plane. (a) White dots indicate location of Rh atom after > 240 cycles at ~ 255 K. (b) After field evaporation of one layer, Ir atom is revealed below the atom sites shown in (a) (after Kellogg [2]).

inserted iridium of 0.76 eV, assuming Do = 1 × 10–3 cm2/sec. At higher temperatures, the rhodium detached from the impurity and self-diffused over the surface with an activation energy of 0.83 ± 0.05 eV, reasonably close to Ayrault’s [3] value. Kellogg also determined the energy for a rhodium atom to dissociate from a substitutional iridium as 0.95 ± 0.02 eV, assuming the same prefactor as for diffusion around the iridium trap. Exchange of iron atoms with gold on the Au(100)-(26 × 5) surface, described as hex, was observed by Herna´n et al. [4] in 1998 using STM. They were able to directly image the Fe atoms embedded into the first Au layer at room temperature and established that the distribution was random, probably due to Fe-Fe repulsive interactions, given that Fe atoms are immobile at room temperature. Based on their STM observations and on Monte Carlo simulations, Herna´n et al. concluded that the Au atoms expelled during the exchange process moved almost in the same way as on a clean Au(100) surface. However, if Fe atoms were embedded into the corners of a reconstructed cell, they acted as preferential sites for nucleation. The effect of gases, specifically adsorbed hydrogen, on the diffusivity of individual adatoms has been examined a number of times. In 1980 Casanova and Tsong [5] looked at how hydrogen affects the self-diffusion of tungsten atoms on the (321) plane of tungsten. Their results are shown in Fig. 6.4 – hydrogen, presumably adsorbed as atoms, inhibited the diffusion. The diffusion barrier was increased by 0.05 eV, but this is probably within the limit of error for the measurements. Nevertheless, based on Fig. 6.4, there is little doubt that the tungsten atom mobility was decreased. A more detailed study of the effect of hydrogen on self-diffusion of nickel atoms on a nickel crystal was carried out at essentially the same time by Tung and Graham [6]. Among other things they compared a nickel tip prepared by thermal annealing with one that had been field evaporated in hydrogen. The differences in the diffusivity on Ni(110) are shown in Fig. 3.46. After hydrogen treatment of the tip, atom diffusion set in at a temperature ~60 K below that on a surface not exposed to hydrogen. All these results were obtained after heating and field evaporation of the tip, which eliminated atom diffusion at cryogenic temperatures at and below 30 K, observed on (110), (331), and (311) planes when hydrogen was still present. Although an understanding of how hydrogen promotes diffusion of nickel atoms, especially on Ni(110), does not emerge from this work, it is clear that there is a very major effect.

426

Diffusion in special environments

/t

1

Clean

Hydrogen

0.1

3.0

Fig. 6.4

3.1

3.2

3.3 1000/T

3.4

3.5

Arrhenius plots for self-diffusion of W adatom on W(321) plane. Diffusion on clean surface indicated by dashed line, on hydrogen saturated surface by solid line (after Casanova and Tsong [5]).

The influence of low coverages of hydrogen impurities on the movement of Ag atoms on the Pt(111) surface was investigated in 1995 by Blandin and Ballone [7] using molecular dynamics with the embedded atom method. They deposited Ag and hydrogen at random positions on the surface, and showed that the presence of hydrogen increased the barrier for Ag movement by around 0.1 eV. Ag and H “dynamically” form a dimer on the surface with a characteristic lifetime around 0.4 psec. However, H embedded between the first and second layer of Pt did not influence the movement of Ag atoms. A year later, a theoretical discussion of the effect of hydrogen on surface diffusion has been given by Stumpf [8], who relied on density-functional calculations for selfdiffusion of Be(0001). He established that hydrogen atoms sat on top of beryllium adatoms and formed a strong bond which weakened the binding of the adatom to the substrate, known as the sky-hook mechanism. This lowered the barrier for surface diffusion from 0.06 eV on the bare surface to 0.02 eV when interacting with hydrogen. The effect of hydrogen on self-diffusion on Rh(100), and also on Rh(311), has been looked at the next year in field ion microscopic observations by Kellogg [9]. The effect, as shown in Fig. 6.5, is an apparent reduction in the diffusion barrier by the hydrogen. This is only apparent, however, as raising the temperature now not only changed the rate but also the hydrogen coverage on the surface. Nevertheless, it is clear that rhodium selfdiffusion occurs more rapidly and at lower temperatures. The effect of hydrogen on Rh(311) is similar, increasing the rate of diffusion. A clear indication of changes in the mean-square displacement for different hydrogen coverages was presented by Kellogg [9] but he claimed that the sky-hook mechanism, a decrease of metal-to-metal bonding through formation of strong metal-hydrogen bonds, was not adequate to explain his finding. The same year [10] Kellogg looked at the effect of hydrogen on self-diffusion on

427

6.1 Near impurities

In [/t](/t in cm2/sec)

–38 Rh on Rh(100) –39

–41 Clean –42 –43 3.0

Fig. 6.5

H2-Exposed

–40

3.2

3.4 3.6 3.8 1000/T (K–1)

4.0

4.2

Arrhenius plot for self-diffusion on Rh(100), for clean and H2 exposed surface (after Kellogg [9]).

c(2x2)

[001]

c(2x2)

(a)

T = 175 K

(b)

(1x1)

c(2x2)

(c) Fig. 6.6

T = 230 K

T = 195 K

(d)

T = 250 K

Sites occupied by Pt atom diffusing on Pt(100) plane at different temperatures and H2 pressures. (a)–(c) c(2 × 2) maps indicate exchange as the diffusion mechanism. (d) (1 × 1) map shows that Pt atom hopping is now dominant. H2 pressures in Torr: (a) 0 (b) 1 × 10–10 (c) 3 × 10–10 (d) 1 × 10–9 (after Kellogg [10]).

the Pt(100) surface. On the clean surface, movement proceeded by exchange, but with a hydrogen partial pressure of ~1 × 10–10 Torr the diffusion rate decreased by three orders of magnitude over 2–3 hours. Exposing the surface to hydrogen in the range 1 × 10–9 Torr at 200 K completely stopped diffusion of the Pt atom on the surface. Kellogg additionally investigated the map of sites occupied by a Pt atom and noticed a change in the mechanism of movement from exchange to hopping, shown in Fig. 6.6. The activation

428

Diffusion in special environments

energy for hopping, assuming a standard prefactor for the diffusivity, was estimated as 0.63–0.66 eV. This study showed that hydrogen does not always increase the mobility of adatoms: slowing down movement seems to be correlated with the mechanism involved in adatom diffusion. Calculations of the effect of hydrogen on self-diffusion on a Ni(100) surface were carried out in the same year with EAM interactions by Haug et al. [11]. They arrived at a diffusion barrier of 0.61 eV on the bare surface. Hydrogen added to the surface interacted attractively (~ 0.02 eV) with a nickel adatom in a nearest-neighbor position, but this attractive force did not change much in the movement of hydrogen adatoms. The presence of hydrogen did, however, change the mechanism of movement of the Ni adatom on the surface. The nickel jumped on top of the hydrogen atom, over a barrier of 0.47 eV, and then to the other side, with an energy barrier of 0.067 eV. Hydrogen was also able to move as a pair together with a nickel atom over a barrier of 0.47 eV. The net effect of hydrogen addition to Ni(100) was to enhance the diffusivity of an adatom. A hydrogen adatom adsorbed close to a Ni island on Ni(100) was also able to destabilize the island by reducing the activation energy for detaching a corner atom. Haug and Jenkins [12] in 2000 continued these investigations, examining Ni(110) and Ni(111) planes. On the former, hydrogen had little influence on movement; the barrier for Ni hopping on the clean surface was 0.39 eV, while in the presence of H it was 0.38 eV. However, the barrier for atom exchange on Ni (110) was lowered from 0.42 eV to 0.38 eV, and now was comparable to the barrier for regular hopping on this plane, raising the possibility that both mechanisms coexist on the surface. On the Ni(111) surface, the movement of atoms between fcc sites required 0.05 eV, while in the presence of hydrogen the barrier was only 0.02 eV. What is interesting is that Ni and H create a dimer when the Ni atom sits in an fcc site while H is in an adjacent hcp site, and the movement of the Ni atom towards (or away from) the H atom is very expensive − 0.07 eV. How hydrogen affects self-diffusion on a Pt(110)-(1 × 2) surface has been probed with an STM by Horch et al. [13]. Addition of hydrogen at a pressure of 3 × 10–8 Torr caused bright spots to appear on the surface at 302 K and platinum adatoms moved more rapidly. At ~5 × 10–7 Torr hydrogen the mean-square displacement was roughly 500 times larger than without hydrogen, and the diffusion barrier was lowered by 0.16 ± 0.02 eV. In separate density-functional calculations it was found that the most stable arrangement was for the hydrogen to be on top of a platinum adatom (the sky-hook mechanism). The situation for an adatom on Pt(110)-(1 × 2) is somewhat complicated, however. The adatom can move along the bottom of the channel, or else it can jump to the (111) walls and continue there, as suggested in Fig. 6.7. On the clean platinum surface the diffusion barrier was estimated at 0.94 eV (somewhat above the experimentally measured value) and movement on the {111} walls was preferred. In the presence of hydrogen the path up the sidewalls becomes more demanding energetically, but along the bottom the barrier is reduced by 0.09 eV, as indicated in Fig. 6.8; this lowering is close to the experimentally found reduction of 0.16 eV. The presence of hydrogen changes the preferred diffusion path from a metastable walk on the sidewalls to a straight movement on the bottom of the channel.

429

6.1 Near impurities

Fig. 6.7

Possible diffusion paths of Pt atoms on Pt(110)-(1 × 2) surface. A atom moves along channel. Alternative paths on the channel walls are possible at fcc and hcp sites. B indicates equilibrium position of Pt-H, with transition state at C (after Horch et al. [13]).

Energy (eV)

1.0

FCC 0.5

0 Fig. 6.8

HCP

HCP

Reaction coordinate

Calculated energy changes for different diffusion paths of Pt on Pt(110)-(1 × 2). Solid curves indicate motion on clean surface, dashed curves are for Pt-H complex. Thick curves give change along direct path (after Horch et al. [13]).

Recently, in 2007, Prévot et al. [14] used SPA-LEED to look at the movement of Au atoms on a Cu(100) surface on which nitrogen atoms had been embedded. They concluded that the embedded nitrogen worked as a trap for moving Au atoms. They fitted the transition temperature between different nucleation regimes obtained from mean-field nucleation theory with kinetic Monte Carlo simulations. Using MC simulations with an embedded array of traps at the surface to model their finding they came out with an energy for migrating Au atoms of 0.4 eV and an energy of traps between 0.2–0.3 eV; these energies did not depend on the size of the traps. In many of the more recent examples, addition of hydrogen has increased the surface diffusivity of adatoms. Only self-diffusion on W(321), Pt(100) and movement of Ag on Pt(111) have shown the opposite behavior, and it will be interesting to test this further. At the moment, this is all the direct information available about diffusion around impurity atoms in the surface, but this is an area interesting enough for additional studies. There are a number of indirect indications of the effect of impurities on diffusion, some of them from island fluctuation measurements others from the creation of “wedding cake” structures, but we will not describe these in detail.

430

Diffusion in special environments

6.2

To descending step edges Even low index surfaces will have occasional lattice steps on them, as it clear from Fig. 6.9 [15], and the question therefore arises how atoms diffuse toward and over such steps. We will first look at the approach of atoms toward descending steps, to find out if adatoms during their descending motion experience only an additional edge barrier Ea exactly at the step edge, as shown in Fig. 6.10, or if the influence of steps has a longer range. We also will explore the mechanism of movement and its dependence upon the chemical identity of the adatoms. In this book we will describe steps by the direction in which they are oriented. For example a step on the fcc(100) surface means a step along the close packed direction . For fcc(111) steps we will use a notation A and B, where both mean steps in the direction; however, the first creates a {100} plane while the second a {111} plane. On an fcc(110) step means parallel to the channels of the plane, while the step is perpendicular. Investigation of adatom movement over descending steps started to be explored by Ehrlich and Hudda [16] in 1966. Using the FIM they established that (110), (211), and (321) tungsten surfaces were bounded by barriers that reflected tungsten adatoms which in diffusion collided with them. From measurements in the vicinity of 300 K they concluded that an additional barrier Ea just 0.065 eV higher than that surmounted in diffusion in the center of the plane would suffice to account for reflection. This study, however, did not provide a truly quantitative value for the height of the barrier, and showed only that interlayer transport of material was limited by the step edge. The

Fig. 6.9

Image of monoatomic steps on a Pt(111) surface obtained in low energy electron microscope LEEM (after Swiech [15]).

6.2 To descending step edges

Fig. 6.10

431

Atom incorporation into ascending and descending steps. Top: side view of adatoms on a stepped surface. Bottom: Potential energy of adatom, showing additional step-edge barrier Ea.

presence of these reflecting barriers is important, as it impedes incorporation of atoms into a descending step, which in crystal growth had previously been assumed to occur readily. The rate of incorporation ri will be given by ri ¼ ν0B expð
EB =kTÞ;

(6:1)

where ν0B is the kinetic prefactor and EB the height of the step-edge barrier. The origin of this additional step-edge barrier Ea, where Ea = EB − ED, is easy to understand. When an adsorbed atom attempts to cross a descending step, the number of interactions with lattice atoms is decreased. The same year as the work of Hudda and Ehrlich, Schwoebel [17,18] invoked step-edge barriers in his theoretical investigations on transport on stepped surfaces. In the literature the additional barrier Ea for atom descent has quite frequently been named the Ehrlich–Schwoebel barrier. Some years later, Tsong [19] examined diffusion of rhenium on W(321) at one temperature, 332 K, and gave a diffusion barrier of 0.88 eV, in agreement with earlier, more extensive studies by Bassett and Parseley [20]; for escape from the plane he estimated a total activation energy EB of 0.932 eV, just 0.05 eV higher than the activation energy for normal diffusion ED. Attempts to define the step-edge barrier EB more quantitatively were made by Bassett [21,22] and his group in 1975. They examined the rate of escape of Ta, W, Re, Ir, and Pt atoms from a W(110) surface at different temperatures. To their surprise, the activation energy they found for escape from the temperature dependence agreed with that for diffusion within their limit of error; in their analysis, they could not detect any additional barrier to leaping over the step edge, yet they did not lose atoms from the surface. The measured values for the step-edge barrier EB were 0.72 ± 0.10 eV for Ta, 0.89 ± 0.10 eV for W, 0.89 ± 0.10 eV for Re atoms, 0.77 ± 0.10 eV for Ir and less than 0.60 ± 0.05 eV for Pt atoms, while diffusion examined in the same study required 0.77 eV for Ta, 0.86 eV for W, 0.99 eV for Re, 0.77 eV for Ir, and 0.62 eV for Pt. This yielded the additional stepedge barrier Ea of − 0.05eV for Ta, 0.03 for W, − 0.1 eV for Re, 0.0 for Ir, and − 0.02 eV for Pt, all negligibly small quantities.1 1

Bassett [21,22] also calculated step-edge barriers assuming a standard frequency prefactor and taking the mean time spent by an atom in the site adjacent to the step; this yielded for the step-edge barrier EB values 1.37 eV for W, 1.26 eV for Re, 1.14 eV for Pt.

432

Diffusion in special environments

1 EB-W = 0.838 eV EB-Re = 1.077 eV Pb /(1–Pb )

EB-Ir = 1.037 eV

0.1

2.4

Fig. 6.11

2.5

2.6

2.7 2.8 1000/T

2.9

3

3.1

Arrhenius plots for the escape probability of W, Re, and Ir atoms from W(110) plane, as measured by Wang and Tsong [23].

Wang and Tsong [23] took another look at the step-edge barrier on the W(110) plane, and examined the rate of loss for W, Re, and Ir adatoms. The (110) plane of tungsten is not ideal for such studies, as the atoms incorporating into a step are not resolved in the field ion microscope. For tungsten atoms on W(110) they estimated an activation energy in two ways. One, by assuming a prefactor Do = 3.6 × 10–3cm2/sec for the diffusivity, giving an increase to the barrier of Ea = 0.228 ± 0.009 eV, which ends up as an overall barrier EB of 1.116 eV. The other, by assuming a frequency factor of vOB = kT/h; in this way they obtained an increase of Ea = 0.199 ± 0.009 eV, which yields an overall activation energy EB of 1.087 eV. For Re and Ir atoms on W(110) they estimated an activation energy only using a frequency factor vOB = kT/h. For the temperature range 390–410 K they found for Re an increase Ea = 0.145 ± 0.006 eV, which gives a total EB of 1.135 eV. For Ir atoms in the temperature range 330–350 K, they arrived at an additional barrier Ea = 0.195 ± 0.012 eV, or a total escape barrier EB of 0.935 eV. These values were obtained from the formula Pb =ð1
Pb Þ ¼ ðnout =nin Þgτν0B exp½
ðED þ Ea Þ=kT;

(6:2)

here nout and nin give the number of paths for an adatom going over the boundary or else inward, and g is a geometrical factor. The authors claimed to obtain erratic results for the activation energy and the prefactor from an Arrhenius plot. However, such a plot for W as well as Re and Ir atoms is shown in Fig. 6.11, and seems quite reasonable. The total activation energy for tungsten is just a little lower than for self-diffusion, for Re we obtain a slightly higher value (Ea = 0.087 eV) than arrived at in this study, and for Ir a barrier still larger (Ea = 0.297 eV). This is similar to what has been reported by Bassett [21,22], and leaves the subject in uncertainty. Fink and Ehrlich [24] in 1984 carried out an interesting exploration of the free energy of the sites on a (211) plane of tungsten. On this surface, shown in Fig. 3.1, self-diffusion occurs entirely along the channels, and by determining the number of times an

6.2 To descending step edges

Fig. 6.12

433

FIM images of Ir atom incorporation into descending step of Ir12 cluster [25]. (a) Ir cluster on Ir(111). (b) Ir atom deposited on top of cluster dominates image. (c) After heating to 120 K, iridium atom is incorporated into cluster.

atom is observed at the different sites their relative free energy can be ascertained. The results for tungsten adatoms were shown in Fig. 3.29. Atoms are fairly uniformly distributed over sites in the center, but the occupation increases dramatically close to the ends of the channel. The free energy of the sites, in Fig. 3.30, shows the expected effects – a significant diminution at the ends, with a small high energy hump on the side toward the (111) plane. For rhenium atoms the binding is somewhat different. At the last position on the plane the free energy is high, with a sharp drop for the next site closer in. Strong binding close to the plane ends is obviously important when an atom attempts to diffuse away from a surface over the step, but in these experiments no attempts to study atom escape over the step-edge barrier were made. However, based on this study, it can be expected that a step on the W(211) plane will influence the descending motion of an adatom not only at the step edge, but also 2–3 spaces away from the edge. A closer examination of atom behavior on top of clusters was done by Wang and Ehrlich [25] in 1991. They found that on a small cluster, made up of 12 iridium atoms and shown in Fig. 6.12, an iridium adatom diffuses at a temperature ~ 20 K higher than on a large plane. However, when the atom reached the cluster edge, it incorporated into the edge at the temperature of migration. This observation suggested no extra barrier for an atom descending from a cluster of size 12. Wang, however, continued his investigations with bigger clusters and found that on larger clusters of 18 or more atoms, self-diffusion occurred as on a large (111) surface, at ~ 100 K. When an atom comes to the edge of the cluster, however, it was trapped there. On warming to T ≥ 150 K the atom escaped over the edge in preference to returning to the center. A crude diagram of the potential energy close to the step deduced from these data is given in Fig. 6.13. It has already been noted that iridium clusters on Ir(111) are bounded by facets of two different structures. As is apparent from Fig. 6.13a, edges of type A are made up of {100} facets, type B edges have a {111} structure, and their properties are different. Incorporation of an iridium adatom occurs at B-type edges by heating to ~ 160 K, as in Fig 6.14c; to incorporate an atom at an A-type edge requires ~175 K, as in Fig. 6.14b. The mechanism of escape is also different at the two edges. That has been tested with the incorporation of a tungsten atom. Incorporation may occur as the result of an atom jumping over the edge and attaching itself to the lattice step, as illustrated in Fig. 6.15a. An alternative, however, is for the adatom to exchange with a lattice atom, as in Fig. 6.15b, leaving the adatom buried in the cluster and pushing out a lattice atom.

434

Diffusion in special environments

(b) Energy

(a)

A B Step

Cluster

Energy

Terrace

[011]

[211]

x

Fig. 6.13

Ir cluster on Ir(111) plane [25]. (a) Schematic of Ir50, showing A- and B-type edges. (b) Schematic of potential energy at step edge. Top: traditional view. Bottom: Results consistent with experiments.

Fig. 6.14

FIM images of incorporation of Ir atom into Ir50 cluster [25]. (a) Initially bare cluster. (b) FIM image after heating cluster with adatom on it to ~ 175 K, showing atom incorporation at A-type step. (c) Atom incorporation at B-type step after warming at ~ 160 K.

Fig. 6.15

Schematic of possible processes for atom incorporation into lattice steps [25]. (a) By atom jump into descending step. (b) By exchange of adatom with cluster atom.

Incorporation of a foreign atom such as tungsten makes it possible to distinguish between the two processes. If a simple jump gives incorporation, a tungsten atom can be detected attached to the cluster. On the other hand, atom exchange will leave an iridium atom attached, and the two alternatives can be easily discerned using FIM since the field to remove an iridium atom attached to a cluster is ~5% higher than for a tungsten atom. Incorporation has been tested, as shown in Fig. 6.16, by depositing a tungsten atom on a cluster of 33 iridium atoms formed on the Ir(111) surface. On warming to 220 K, the atom attaches to a site of type A, and is removed by a low voltage of 9.2 kV. Incorporation into a B-type step, shown in Fig. 6.17, is examined in the same way. To remove the atom

6.2 To descending step edges

435

Fig. 6.16

FIM images of incorporation of tungsten atom into A-type step of Ir33 [25]. (a) Ir cluster, on top of which Ir atom will be deposited. (b) After heating to 220 K, atom incorporates into A-type step. (c) Protruding atom has been field evaporated at 9.2 kV, typical of tungsten.

Fig. 6.17

FIM images of tungsten atom incorporation into step of Ir69 cluster [25]. (a) Field evaporation has created cluster on Ir(111), and W atom was then deposited on it. (b) After heating to 220 K, atom has incorporated into B-type step. (c) Field evaporation removes atom at 9.8 kV, typical of Ir atom.

attached to the step now requires a higher voltage of 9.8 kV, indicating an iridium atom, not tungsten, has been removed. Such observations are a direct proof that at a B-type step incorporation occurs by atom exchange, while at an A-type step a simple hop is involved. The mechanism seems to be nicely correlated with the atomic structure of edges since step A has an atom lying directly in front of every edge atom, which might block exchange, while step B provides a channel, which probably constitutes an easy path for movement of the edge atom. A similar situation was observed for Re on the Ir(111) surface [26]. Descent at a B-type step proceeded here also by exchange; when incorporation took place at a step of type A it occurred by simple hopping. The work of Wang and Ehrlich showed that the descending movement of an adatom is affected not only by the additional energy barrier at the step edge, but also by the mechanism of movement. Up to this point, experimental information about step-edge barriers had been derived from observations on fairly high melting materials. Interest in crystal growth had constantly grown, however, and there clearly was a need for data on step-edge barriers of metals like copper or silver, not readily susceptible to FIM analysis. Models therefore began to be developed to derive this information directly from growth experiments, as second layer nucleation on islands is affected by the size of the step-edge barrier. Theoretical and experimental investigations started blooming and for clarity of the presentation we therefore divided further findings according to the material, in the order they appear in the periodic table.

436

Diffusion in special environments

total energy (eV)

–2.7 –3.0 –3.3 –3.0 –3.3 –3.6 –3.9

1 2

–10 –5 0 5 10 15 generalized coordinate (bohr) Fig. 6.18

Energy along diffusion path of Al atom at a {111} step on Al(111). Upper curve for hopping, lower one for atom exchange (after Stumpf and Scheffler [28]).

6.2.1

Aluminum The energetics of atom incorporation from the top of an aluminum cluster were investigated by Stumpf and Scheffler [27–29] in 1994, using density-functional theory and the local-density approximation. Their estimates for the total energy of an aluminum adatom on the Al(111) plane with B-type {111} edges are shown in Fig. 6.18. For hopping over the edge the barrier EB was found as 0.33 eV, but exchange required only 0.06 eVand was clearly favored for incorporation. For step A the energy EB required for hopping amounted to 0.45 eV, compared with an energy of 0.08 eV for exchange. Both steps involve an additional barrier Ea for descent, since diffusion on the terrace requires only 0.04 eV. It is interesting to note that the adatom energy at the edge is 0.15 eV lower than in the plane center, which is similar to effects found earlier in experiments on Ir(111). The authors ruled out electrostatic attraction of adatoms and step dipoles as well as elastic interactions of adsorbate-induced and step-induced relaxation as the explanation. They speculated that it can be caused by interaction of adatoms and step-induced surface states. Diffusion of dimers and trimers down an Al(111) surface was examined by Bockstedte et al. [30] using the EAM potentials of Ercolessi and Adams [31]. Both dimers as well as trimers dissociate during descent, and for both the descent proceeds by exchange. The descent of a dimer down {111} steps (B steps) can be divided into two processes: first the whole dimer moves down by exchange of one atom, the dimer bond remaining unchanged. Then the dimer dissociates as one atom is incorporated into the step, while the second atom stays on the island but may descend later by exchange, not necessarily at the same place. The energies associated with these two processes are 0.1 eV and 0.08 eV respectively, as illustrated in Fig. 6.19. If the remaining atom diffuses down the step following the first one such a transition requires an energy of 0.11 eV. Diffusion of trimers down a B step proceeds in analogous fashion. After the trimer approaches the step, it starts moving down by exchange. Dissociation occurs and the remaining dimer now moves over the surface. The process is illustrated in Fig. 6.20a. The energy barrier associated with the movement of the trimer by exchange involves

437

6.2 To descending step edges

1.60 1.55 1.50 0.10eV 0.08eV

Energy (eV)

1.45 1.40 1.35 1.30 1.25 1.20 1.15

0.0

0.1

0.2

0.3

0.4

0.5

Fig. 6.19

Potential energy calculated with Ercolessi and Adams potential [31] in diffusion of Al dimers down {111} faceted step of Al(111) surface (after Bockstedte et al. [30]).

Fig. 6.20

Potential energy in diffusion of Al3 down {111} faceted step. (a) For diffusion of first atom. (b) Diffusion of second atom (after Bockstedte et al. [30]).

overcoming two barriers, 0.10 eV and then 0.12 eV for dissociation. The descent of the remaining dimer next to a newly created kink costs 0.26 eV and is shown in Fig. 6.20b. Migration of the last trimer atom away from the kink occurs over a barrier of 0.16 eV. Movement down {100} A-type steps is much more energetically demanding. For a dimer the descent over this step costs 0.39 eV, for a trimer 0.38 eV is needed for the first part and 0.13 eV to continue the process. Descent of the last atom in the dimer requires 0.15 eV, and in the trimer 0.16 eV. Activation energies at B-type steps as well as at A-type {100} facets are both listed for dimers and trimers in Table 6.1. It is clear that the initial step of going over the edge occurs much more easily at B-type edges for both dimers and trimers.

438

Diffusion in special environments

Table 6.1 Energetics of an aluminium cluster diffusing down a (111) island [30]. Diffusion barrier EB (eV)

Fig. 6.21

Facet

A-type

B-type

Dimer Trimer (process 1) Trimer (process 2) Last atom in the dimer Last atom in the trimer

0.39 0.38 0.13 0.15 0.16

0.10/0.08 0.10/0.12 0.26 0.11 0.16

Atom motion and energetics for Pb ascending monatomic height step on Al(110). (a) In-channel jump. (b) Cross-channel jump (after Zhu et al. [33]).

In 2002 Liu et al.[32] used molecular statics to investigate descending motion over multilayer steps, a subject discussed in Section 6.5. For comparison they also derived energetics for descending a monolayer step. Descending an Al(111) surface step along the direction an energy EB of 0.06 eV for step type B and 0.30 for step type A were required. While descending from Al(111) over a /{110} step2 required the same energy as descending down step B. Descending from Al(100) over a /{100} step required an energy EB of 0.35 eV, while over a /{110} step required an energy EB of 0.34 eV; stepping down a /{111} step is more demanding – EB = 0.45eV. On Al(110) descent over a /{110} step needed EB = 0.33 eV, while descents over a /{100} step perpendicular to the channels of the plane and a /{111} step parallel to the channels of the plane were more demanding, with EB of 0.63 and 0.7 eV respectively. All these transitions were accomplished by atom exchange. Liu et al. did not derive energetics for terrace movement in this study. Aspects of diffusion on Al(110) have also been examined in density-functional calculations by Zhu et al.[33], who looked at atom ascent and descent at steps. Two different geometries, shown in Fig. 6.21, were considered, in which the adatom exchanges with a lattice atom. For ascent in an aluminum channel, an aluminum atom was found to require 0.60 eV; the downward step involved a barrier of only 0.43 eV. An 2

Step /{110} means the step position is along the direction with the structure of the {110} plane.

439

6.2 To descending step edges

upward move in the cross-channel direction entailed a barrier of 0.67 eV, while 0.71 eV for going down. No experimental data are available for descent down aluminum steps, and the theoretical information available is unfortunately also limited. However, from existing data the most likely mechanism of descent will be exchange, not hopping, for all types of edges.

6.2.2

Iron Information about the step-edge barrier on the iron(100) surface is provided by the STM experimental observations of island fluctuations by Stroscio et al. [34]. The first analytical analysis based on these data was done by Smilauer and Harris [35] who were able to deduce an additional step-edge barrier Ea in the range 0.0 eV to 0.10 eV. Estimates were continued by Amar and Family [36], who based their work on the experimentally measured diffusion barrier ED of 0.454 eV from Stroscio et al. [34]. They then looked at the RHEED intensity and compared it with Monte Carlo simulations using different barriers for incorporation into an edge site. With the STM findings of Stroscio et al. at room temperature they found an additional step-edge barrier Ea between 0.03 eV and 0.06 eV, as indicated in Fig. 6.22. The mechanism of atom descent was not determined. In 1999, Bartelt and Evans [37] simulated the epitaxial growth of Fe on Fe(100) and compared these with measurements carried out with the STM, again by Stroscio et al. [34]. In this way they came up with a step-edge barrier Ea of 0.03 eV to 0.04 eV, values in agreement with the earlier findings of Amar and Family [36]. The epitaxial growth of iron on Fe(110) was examined by Köhler et al. [38] with a scanning tunneling microscope, and the atomistic results were then compared with Monte Carlo simulations to give the following additional step barriers Ea for transitions in the ½1
11 direction: edge, 0.41 eV; edge, 0.34 eV and edge,

1.5

RHEED Intensity (arb. units)

1.5 (b) 20 °C 1

1 IR

20 °C 180 °C 180 °C

0.5

0.5

250 °C

0

Fig. 6.22

(a)

Fe/Fe(100)

0

1

2

3

4

5 0 1 2 Coverage (monolayer)

250 °C

3

4

5

6

0

Diffusion of Fe atoms on Fe(100) surface. (a) RHEED intensity data at different temperatures. (b) Simulation with step barrier of 0.03 eV (solid line) and without step barrier (dashed line) (after Amar and Family [36]).

440

Diffusion in special environments

0.36 eV. The anisotropy of the step-edge barriers agrees very nicely with the elongated shape of the islands observed with STM. There are no theoretical insights into the step-edge barriers of iron surfaces, and experimental findings are also limited to one measurement on Fe(100) and one on Fe(110).

6.2.3

Nickel Adatom descent over nickel as well as copper steps have received quite wide theoretical attention, unfortunately with a complete lack of experimental findings for comparison. Calculations started in 1992 with the work of Liu and Adams [39], who used the embedded atom method to calculate activation energies for a Ni atom descending at Ni steps. They did not observe any exchange; however they did not clearly identify the kind of step for the Ni(111) surface. The highest activation energy for descent was observed on Ni(110), EB = 1.34 eV, where they looked at the steps along the channels of the plane. This value should be compared with the movement along the channels on the terrace, which required only ED = 0.44 eV. The second highest energy for descent was for Ni(100) over a step, where Liu and Adams found a barrier EB of 0.91 eV, while diffusion on the terrace cost only 0.63 eV, and the lowest, on Ni(111), was EB = 0.55 eV; diffusion on the (111) terrace required only 0.056 eV. The fact that a jump instead of exchange occurred suggests a step of type A rather than type B. We can also see that the order of the energies needed for descent is not the same as for movement on the terraces. Liu and Adams [40] continued their work using EAM and molecular statics. They found a forbidden region next to both ascending and descending steps, with an interior barrier 2 to 3 spaces away from the step on Ni(111). The diffusion barrier had a constant value of 0.056 eV, as long as the atom is a distance of > 5 spacings away from the step. Approaching the descending step the barrier rose to 0.064 eV, and then the adatom experienced lower barriers than for movement on the terrace. When the adatom reached the step the barrier rose again. It was also shown that on a B-type step descending by exchange is more likely than by a jump over the step, the barrier EB amounting to 0.169 eV compared with 0.505 eV for jumps. For a type A step, the energetics for exchange and jumps were comparable, EB being in the range of 0.55 eV compared with 0.585 eV. Descending from Ni(110) at a step and from Ni(100) by a step proceeded by exchange with an energy EB of 1.28 and 0.85 eV. For jumping over steps they found the same values as in their previous study. They did not observe any forbidden regions on the latter planes. The movement on the terrace occurs over a barrier of 0.41 eV for in-channel diffusion and 0.49 for cross-channel diffusion. On Ni(100), the descending barrier by exchange was 0.22 eV higher than for movement on the terrace. The next investigations of Ni steps were done in 1994 by Stoltze [41], who looked briefly on atom movement over step edges using effective medium theory. For Ni(100) over steps he found a value EB of 0.787 eV for hopping compared with 0.890 eV for exchange. In his investigation, movement on the terrace required 0.558 eV. For Ni(110) he looked at movement over steps (across channels), for which jumping cost EB = 1.178 eV while exchange required EB = 0.917 eV. An adatom descending along channels of Ni(110) cost EB = 0.68 eV, while exchange was more demanding, needing

441

6.2 To descending step edges

Table 6.2 Comparison of energy barriers (eV) for diffusion of fcc metals from fcc(111) calculated with CEM [42].

ED (Jump) EB (Ex) EK (Ex)

Ag

Cu

Au

Pd

Ni

Pt

0.020 0.064 0.030

0.039 0.176 0.122

0.029 0.090 0.047

0.034 0.030 0.005

0.036 0.094 0.067

0.048 0.044 0.012

EB = 0.81 eV. Terrace movement on Ni(110) involved a barrier of 0.407 eV for the inchannel direction and 1.157 eV across channels. On Ni(111) jumps over the edge cost EB = 0.529 eV, exchange EB = 0.669 eV. A- and B-type steps were not recognized, and movement on the (111) terrace cost only 0.068 eV. Li and DePristo [42,43] undertook an effort to better understand homoepitaxy on fcc(111) surfaces by calculating barriers to surface diffusion compared to atom incorporation into lattice steps by exchange. For this they employed the corrected effective medium theory, CEM, and obtained the barriers to surface diffusion over B-type step edges listed in Table 6.2. For Ni(111) the barrier for descending was EB = 0.094 eV, while descending at a kink was easier, requiring EK = 0.067 eV. To overcome the step-edge barrier, an adatom needs a higher energy than for diffusion on the flat surface, for which the possibility of exchange was not taken into account. It is clear after examining the data in Chapter 5 that some of these estimates are way off. In fact, the authors pointed out that theory was not designed for quantitative data and that only relative information was valid. Also examined were the interlayer diffusion processes. For small clusters of seven atoms or less the barrier to incorporation was very high, but decreased by an order of magnitude once the cluster size exceeded ten atoms and then was almost size independent. This is not what has been found in the few experiments done, however. Although the results of Li and DePristo’s work should not be taken too seriously, they do show that for palladium, nickel, and platinum the incorporation barrier is smaller for nineteen atom clusters than at kinks in the steps. In 1997 Merikoski et al. [44] looked at step characteristics on fcc(100) planes. For the Ni(100) surface descending by exchange over a step seemed to be more probable, with a barrier EB of 0.744 eV compared to 0.920 eV for jumps. Movement on the terrace proceeded by jumping which required only 0.631 eV compared to 0.844 eV for exchange. More interesting are the findings of Merikoski et al. for kinks. For Ni the value of EK for exchange at kink sites was a bit lower than for hopping, 0.522 eV for exchange and 0.695 eV for jumps. The barrier for exchange at the kink was lower than for hopping on the flat plane. For all the three materials they investigated, the activation energy for descending into kinks is lower than the energy of descending at a straight step. The influence of hydrogen on atom descent on nickel surfaces was investigated by Haug and Jenkins [12] using EAM interactions. For descending a straight step by jumps on Ni(100) without any hydrogen, the energy barrier EB was 0.95 eV, leaving an additional step-edge barrier Ea in the range of 0.34 eV; descending at a kink was a bit

442

Diffusion in special environments

easier, requiring EK = 0.7 eV or an additional barrier of 0.09 eV. Hydrogen promoted the descending motion at both straight and kinked steps. On the former the barrier for descending with H present was 0.28 eV lower than without, while at the kink the barrier was 0.09 eV lower. However, the movement on the flat surface in the presence of H was also easier, 0.47 eV compared with 0.61 eV in the absence of H. For Ni(110) surfaces, descent by adatom exchange along the short axis, along the channels, cost EB = 0.69 eV on the clean surface, and almost the same, 0.68 eV, in the presence of H. Stepping down by jumps on the clean surface cost 1.33 eV along the long axis, across the channels, and 0.75 eValong the short. The barrier for jumping was lowered by the presence of hydrogen to 1.05 eV across the channels and 0.73 eV along them, but jumps were still more demanding than exchange. On this plane, hydrogen does not seem to play a big role in interlayer transport. Terrace diffusion also does not feel the influence of hydrogen: diffusion occurs over a barrier of 0.39 eV on the clean surface compared with 0.38 eV on the surface with hydrogen. Hydrogen, however, does seem to influence movement on Ni(111). At an A-type step, descent by exchange without H cost EB = 0.23 eV, while with H only 0.18 eV. These values can be compared with movement on the terrace, which occurs over a barrier of 0.05 eV without H and 0.02 eV with H present. For type B steps the barriers were in the same range as for A-type steps. For Ni(111) steps the mechanism of descent was not clearly deduced. The study of Liu and Adams [40] slightly prefers exchange over atom jumps for both types of edges, but their barrier is much higher than obtained later for jumping by Haug and Jenkins [12], leaving the situation confused. On Ni(100), two studies favor exchange over jumps while one indicates jumping over exchange, the rest do not consider both mechanisms for comparison. However, kink sites seem to provide an easier descending path. There is also disagreement between the values derived for the Ni(110) surface.

6.2.4

Copper(111): experiment The first attempt to derive the barrier for interlayer movement on Cu(111) based on experimental findings was done by Markov [45] who arrived at the critical temperature for the transition from crystal growth by step flow to two-dimensional nucleation. Using experimental data from diffraction studies, he was able to give the additional step-edge barriers. He relied on experimental findings of van der Vegt et al. [46] to arrive at a value of Ea = 0.113 eV. From experiments of Wulfhekel et al. [47] a slightly different value of Ea = 0.080 eV to 0.055 eV was derived. Icking-Konert et al. [48] looked at the decay of Cu adatom islands on Cu(111) in STM experiments at varying temperatures. They arrived at a value of Ea = 0.116 ± 0.002 eV for the additional step-edge barrier by comparing experimental findings and simulations. A year later, the same group [49,50] analyzed the decay of a 40 multilayer Cu island on Cu(111) and found a step-edge barrier Ea of 0.224 ± 0.009 eV, as shown in Fig. 6.23. The pre-exponential factor for hopping over the step was in the range 3.5 × 1012 ± 1 sec−1. This is the weighted average over all possible crossing processes at A- and B-type steps. They also found that the value of the step-edge barrier was independent of terrace width, as long as this was larger than 14 ± 2Å. The authors reanalyzed results from a previous paper

443

6.2 To descending step edges

s(T ) = s0 exp(–EB /kT )

0.1 Cu(111) Ea = 0.224 ± 0.009 eV s0 = 3.5 × 10±0.35

0.01

0.001

Fig. 6.23

2.6

2.8

3.0 3.2 1/T (10–3K–1)

3.4

Arrhenius plot for decay of multilayer Cu(111) island obtained by fitting STM data to rate equations, yielding a step-edge barrier of 0.224 ± 0.009 eV (after Giesen and Ibach [50]).

[48] with a “sticking coefficient” prefactor s0 = 3.5 × 10±0.35and came out with a value of Ea = 0.19 ± 0.02 for the additional step-edge barrier [50]. Diffusion of copper atoms on Cu(111) and Cu(100) was looked at by Ferrón et al. [51], who relied on a second-moment approximation to tight-binding theory [52]. From He scattering data they concluded that Cu(111) had poor interlayer transport; for Cu(100), however, layer-by-layer growth was observed, suggesting that on this surface good interlayer mass transport prevailed, as indicated in Fig. 6.24. From Monte Carlo simulations Ferrón et al. established that diffusion over the flat Cu(111) plane occurred by hopping. At both A- and B-type edges, however, the atoms go to the lower level by exchange with an atom from the lattice, as shown in Fig. 6.25. Exchange descent was also suggested for the Cu(100) plane. The additional step barrier Ea was calculated as 0.03 eV on Cu(100), and 0.057 eV for Cu(111). Differences between descent at the two kinds of steps were not investigated. They also looked at the presence of 1ML of Pb on Cu(111) and found that this surfactant changed the mechanism of atom movement on the surface from hopping to exchange, and suppressed the step-edge barrier, facilitating layer-bylayer growth. What is more surprising was that movement of Cu proceeded below the Pb layer.

6.2.5

Copper(111): theory The investigations of interlayer movement for copper surfaces started with the study of Stoltze and Nrskov [53]. They carried out effective medium theory calculations for diffusion and incorporation into a lattice step from above for copper adatoms on copper clusters of three and seven atoms with the Cu(111) structure. Estimates were always made for B-type edges. For adatom jumping over the edge on the trimer they found an activation energy EB of 0.284 eV, compared to 0.230 eV for exchange, and on the heptamer these figures were 0.418 eVand 0.267 eV for exchange. Diffusion on the cluster required overcoming a barrier of 0.093 eV, revealing a considerable energy requirement

444

Diffusion in special environments

Fig. 6.24

Thermal energy scattering showing homoepitaxial Cu growth. (a) Interlayer diffusion is suppressed on Cu(111), giving multilayer growth. This is shown by monotonic decrease of specular intensity and by STM image. (b) Film grows easily on Cu(100) surface (after Ferrón et al. [51]).

for incorporation. This work was presented at a meeting in 1991, and stimulated the experiments by Wang and Ehrlich [25]. Based on their own findings, Stoltze and Nrskov also postulated that the barrier for hopping over the edge was very size dependent, while exchange did not seem to be so dependent, findings opposite to what was seen in theoretical investigations on flat surfaces. Stoltze [41] continued working with EMT and for Cu(111) steps, an adatom jump cost only EB = 0.365 eV and is more probable than exchange, which required 0.45 eV. Again going over the step was much more demanding than movement on the flat plane, which occurred over 0.053 eV. Karimi et al. [54] utilized EAM potentials to derive the activation energy of diffusion ED for an adatom as 0.028 eV; movement from an upper to a lower level occurred over a barrier EB of 0.085 eV by exchange with a lattice atom, compared to 0.49 eV for a jump. Only steps of type B were investigated. For all steps investigated, atom descent by exchange was energetically favorable. They also examined diffusion of atoms near the steps. Movement along steps required similar energies of 0.29 eV, movement perpendicular to the ledge needed 0.67 eV for Cu(111). Breeman [55] used the Finnis–Sinclair potential [56] and found that the movement from fcc to hcp sites required 0.13 eV, and for the reverse the barrier was 0.11 eV. Interlayer step diffusion at a {111} step (B-type) cost EB = 0.27 eV for exchange and 0.53 eV for hopping. Descent from {100} steps (A-type) by a jump was comparable to exchange, EB = 0.53 eV for jump and 0.56 eV for exchange. Li and DePristo [42,43] calculated barriers to surface diffusion and compared them to atom incorporation by exchange into lattice steps using the corrected effective medium theory, CEM; the

445

6.2 To descending step edges

Cu(111)

3.0

Coordinate (nm)

x 2.0

1.0 y z 0.0

Cu adatom

0

10 20 MC steps (×10–3)

Cu(111) - [111] step 3.0

Coordinate (nm)

x 2.0

y 1.0

z 0.0

Fig. 6.25

0

10 20 MC steps (×10–3)

30

Cu adatom

Monte Carlo simulations of Cu diffusion on Cu(111). Left top: adatom approaches step and then exchanges with edge atom. Left bottom: time evolution of adatom coordinate. Right top: coordinate of Cu atom diffusing on Cu(111) terrace. Right bottom: adatom trajectory (after Ferrón et al. [51]).

barriers to surface diffusion over B-type step edges are listed in Table 6.2. For Cu(111) they found a barrier equal to EB = 0.176 eV, at the kink EK = 0.122 eV, and the barrier for interlayer movement was much higher than for diffusion on the flat surface. It should be noted that their values are way off from other investigations. They also examined interlayer processes for small clusters of seven atoms or less and found the barrier to incorporation was very high; the barrier decreased by an order of magnitude once the cluster size exceeded ten atoms and then was almost size independent. This is not what has been found in the few experiments done, however. Trushin et al.[57] used EAM to look at the Cu(111) plane, and found that jumps over a {100} faceted step of type A required EB = 0.51 eV, but only 0.28 eV for exchange; at type B steps, {111} faceted, EB = 0.5 eV was needed for jumps and 0.085 eV for exchange. Movement on the flat surface required an energy ED of only 0.029 eV. As we can see, descent on Cu steps seems to proceed by exchange rather than by jumping; additionally descent through step B is much more likely than over step A. On the Cu(111) plane,

446

Diffusion in special environments

descending by exchange close to a kink in an A-type step required EK = 0.2 eV, by a jump 0.4 eV. For a kink in the B-type step, exchange was also more favorable than a jump, with a barrier of EK = 0.31 eV compared with 0.39 eV for the jump. However for B-type steps, kinks did not provided an easy descent, compared to straight steps. The fast island decay observed by Giesen et al. [58] for Cu(111) mounds as well as by Morgenstern et al. [59,60] for Ag(111) triggered theoretical investigations by Larsson [61], who used a Monte Carlo bond-counting model to show that such fast decay of Cu(111) islands can be explained by adatoms hopping over the edge, not by exchange. For crossing an A-type step he derived a barrier EB of 0.24 eV, while for a B-type step it was 0.47 eV. The activation energy for diffusion ED on the terrace was estimated as 0.09 eV. He claimed that enhanced intralayer diffusion at the corners as well as step–step interactions can be responsible for the rapid decay of small islands. Feibelman [62] used ab initio barriers for exchange at the vicinal steps of the Cu(111) plane and found them to be in the range EK = 0.66–0.85 eV when kinks on the terraces were close. In 2003 Huang et al. [63] used EAM potentials in molecular dynamics/statics simulations to look at the multilayer descent from the Cu(111) surface. Monolayer descent to another (111) plane required EB of 0.1 eV over a B-type step and 0.34 eV over a step of type A, but terrace movement on Cu(111) needed only ED = 0.04 eV. In 2004 Wang et al. [64] investigated with EAM the movement of adatoms between facets and for monolayer descent from Cu(111) over steps. For B steps they showed that descent required EB 0.08 eV by exchange and 0.5 eV by hopping. A step exchange required a barrier EB of 0.28 eV, while hopping had the same energy as at the B step – 0.5 eV. Compared with an energy ED of 0.04 eV for hopping on the flat plane, there is clear existence of an additional barrier Ea at both types of steps. Recently Smirnov et al. [65] used ab initio and MC simulations to look at the interactions of a Cu(111) island with adatoms deposited on this island. They found a repulsive interaction of 30 meV at the first row of hollow sites from descending steps A and B. In the second row of hollow sites the interaction was also repulsive, ~10 meV. Attractive minima of − 19 meV (A-type of step) and − 20 meV (B-type of step) were observed at distances 1.5 Å from step A and 2.9 Å from step B. They associated this effect with redistribution of the electron charge-density at the step. Such interactions may cause empty zones (zones where adsorption of adatoms was not noticed) to be observed experimentally on the cluster. Recently the descending motion of iron was investigated by Mo et al. [66] and for movement down a step by exchange they derived a barrier of 0.07 eV; diffusion on the terrace required 0.025 eV. The same system was also investigated by Ding et al. [67]. An oscillatory dependence was observed with the barrier at a descending edge of 0.173 eV; this was for a jump, not by exchange, as in the previous study. Descending motion from B-type of steps on Cu(111) seems to be preferably by exchange and theoretical values for the additional barrier varied from 0.038 to 0.14 eV. The situation for descending over step A is more confusing – half the studies suggest exchange, the other half jumps. Theory however is in agreement that descent over step A is more demanding than descent over step B. Unfortunately experiments so far have derived only indirect measurements averaged over all types of steps.

6.2 To descending step edges

6.2.6

447

Copper(100) There is no experimental investigation of adatom descent from Cu(100) surfaces. Most theoretical studies tackled descending motion over close-packed steps. Investigations where done by Stoltze [41] with effective medium theory. Adatom descent over a step needed an energy EB of 0.572 eV by jumps and 0.626 eV by exchange. This should be compared with 0.425 eV for movement by jumps on the terrace. Liu [68] also used EMT interactions for determining the energy necessary for jumping and exchange over steps. For jumps EB = 0.81 eV was required, while exchange only needed 0.65 eV, and diffusion on the terrace was over a 0.45 eV barrier. Calculational efforts to determine barriers continued, and Karimi et al. [54] made detailed calculations using EAM potentials for self-diffusion of copper atoms on copper. At steps diffusion by jumps required overcoming a barrier of EB = 0.77 eV, but moving over an edge by atom exchange took an energy EB of 0.51 eV. Movement on the flat surface cost 0.48 eV. Movement along steps by hops required 0.25 eV while perpendicular to the ledge 0.84 eV. In 1996 Breeman et al. [55] looked at diffusion on the Cu(100) plane with the Finnis– Sinclair potential [56] with and without indium present. A copper adatom jumping over the step edge required EB = 0.51 eV, jumping next to a kink required EK = 0.4 eV. Exchange at the kink was comparable with jumping over a straight edge, costing 0.52 eV. Exchange with an indium atom embedded into a straight step cost only 0.26 eV, exchange with an In atom embedded in the kink site cost even less, 0.16 eV. Breeman did not derive the energetics for movement of Cu on the flat (100) surface, that is why it is difficult to know if there was an additional barrier at the step edge. Trushin et al. [57] also looked at copper surfaces with EAM. On a flat Cu(100), atom jumps occurred over an activation energy ED of 0.49 eV and of 0.69 eV for exchange. Descent over a step required EB = 0.33 eV for exchange and 0.57 eV for jumps, so the barrier for exchange over this step is lower than the barrier for motion on the flat terrace. For the close-packed step of this plane the barriers are a bit higher, EB = 0.54 eV for exchange and 0.77 eV for jumps. The descent in the presence of a kink in the step was also investigated. Exchange with a corner atom of the kink required EK = 0.34 eV, while a jump was more energetic and cost 0.57 eV. Kinks seem to provide the easier way for interlayer motion. However, what is surprising in these investigations is that the descending motion of an atom on Cu(100) by exchange has a lower energy barrier than movement on the flat surface. Unfortunately there are no direct measurements available to support this claim. Additionally, descending over steps (if they exist on the surface) is easier than over steps. In 1997 Merikoski et al. [44] looked at step characteristics on Cu(100) planes. Movement on the flat surface proceeded by hopping over a barrier ED of 0.399 eV. Based on their findings, jumping over the edge was more favorable than exchange, with an activation energy of EB = 0.578 eV compared with 0.631 eV. The favored mechanism of descent for copper does not agree with the previous investigation of Trushin et al. [57] which suggested exchange with a very low energy as the mechanism on this plane. More interesting are the findings of Merikoski et al. for kinks. For both Cu

448

Diffusion in special environments

and Ag, hopping and exchange at kinks seem to have almost the same energy. For Cu, EK = 0.442 eV for jumps compared with 0.422 eV for exchange. For all three materials, including Ni, investigated by Merikoski et al., the activation energy for descending into kinks is lower than the energy of descending at a straight step, and for all three of them the barrier for descending motion was higher than the barrier for movement on the flat surface. Montalenti and Ferrando [69] used RGL potentials to calculate the step-edge barriers on Cu(100). They did not describe the step orientation but most likely they were steps. Descent via jumps required EB = 0.71 eV, while exchange needed only 0.52 eV. Both barriers are higher than for movement over the plane, which required ED = 0.39 eV. Ferrón et al. [51] relied on a second-moment approximation to tight-binding theory [52] and He scattering data. For Cu(100) they observed layer-by-layer growth, suggesting that on this surface good interlayer mass transport prevailed, as indicated in Fig. 6.24. Exchange descent was suggested with the additional step barriers Ea as 0.03 eV. Wang et al. [64] investigated descent from a Cu(100) surface over steps with EAM and found that an energy EB of 0.32 eV was needed for exchange, while hopping required 0.55 eV. Descent over steps required an energy of 0.57 for exchange, 0.81 for hopping. Movement on the flat Cu(100) was shown to proceed by hopping with an energy ED of 0.48 eV. Surprisingly, the value for descent by exchange was lower than for movement on the terrace. Again, descent over a step was easier than over a step, findings in excellent agreement with the results of Trushin et al. [57]. In 2006 Yildirim et al. [70] did calculations using EAM semi-empirical potentials. For jumping over a step they arrived at a barrier EB of 0.79 eV, which indicated an additional step-edge barrier Ea of 0.185 eV. Most of the calculations prefer exchange over hopping as the mechanism of descent from the Cu(100) surface. Movement seems to be easier through rather than steps; however, the latter is more frequently observed on the surface. Studies also suggest easier descent at kinks.

6.2.7

Copper(110) Descent motion from the channeled Cu(110) surface was studied by Stoltze [41] with EMT. Two directions of descent, ½1
10 and [001], from the copper (110) surface were explored. For jumping along the ½1
10 direction down a step, an adatom needed EB = 0.48 eV, while in direction [001] 0.84 eV was required. Exchange is more favorable 10 direcfor the [001] direction, requiring EB = 0.69 eV, and more demanding in the ½1
tion, at 0.574 eV. Overcoming both steps is associated with an additional step-edge barrier, since jumps on the terrace cost 0.292 eV in the ½1
10 direction and 0.826 eV along [100]. Interlayer diffusion on this plane continued to be studied by Karimi et al. [54] using EAM potentials. Diffusion on the flat Cu(110) involved a barrier ED of 0.24 eV, and diffusion down a step by atom exchange required an activation energy EB of 0.72 eV; jumps are unlikely, as they cost 1.17 eV. Movement along a step by jumps required 0.28 eV, perpendicular to the ledge 1.17 eV.

6.2 To descending step edges

449

Diffusion of copper over steps on Cu(110) was probed by Zhu et al. [33] in densityfunctional calculations. An in-channel upward exchange involved a barrier of 0.84 eV, and 0.54 eV downward. To move up in the cross-channel direction, previously shown in Fig. 6.21, required 0.75 eV, but 0.82 eV down. Yildirim et al. [70], using EAM, found a barrier EB = 0.644 eV for jumps over Cu(110) steps perpendicular to the channel, so the additional step-edge barrier Ea was 0.414 eV. Prefactors were all on the order of 10–3cm2/sec. Recently movement of Co adatoms on the Cu(110) was investigated with molecular statics. For descent via exchange down steps a barrier EB of 0.66 eV was derived, while descent over cost 0.60 eV. Movement of Co adatoms on the terrace by exchange required 0.32 eV. The number of theoretical investigations on Cu surfaces is huge compared with a very limited experimental picture. It will be important to have more direct measurements of adatoms overcoming steps, even though for copper, calculations have yielded good results.

6.2.8

Rhodium Very limited information is available in the literature about rhodium steps. The only data come from an investigation of self-diffusion on small islands on Rh(111), examined by Ma´ca et al. [71,72] using RGL interactions between atoms. The activation energy for rhodium self-diffusion on defect free (111) surfaces was found to be 0.15 eV, in very good agreement with the experimental value of 0.16 ± 0.02 eV [3]. For jumps over the straight edges the activation energy EB was found to be similar for the two types: 0.73 eV for Rh at A-type steps and 0.74 at B-type. For exchange at straight steps the barriers EB were considerably smaller, 0.39 eV for B-type steps and 0.47 eV for A-type steps. The presence of kinks seemed to influence descent on Rh; the activation energy EK for exchange at B-type steps reached a low of 0.24 eV. For steps of type A, descent over a kink by exchange was comparable with descent over a straight step. The authors also looked at descent from 3 × 3 islands. For small Rh islands the energy for descending by exchange was much lower for type B steps, 0.24 eV, but a bit higher at 0.43 eV for type A. Rhodium steps definitely require much more scientific attention from both experiment and theory.

6.2.9

Palladium The first investigations of palladium steps were done by Stoltze [41] using effective medium theory. For Pd on Pd(100) overcoming a probable step required an energy EB of 0.573 eV for jumps and 0.698 eV for exchange. These values can be compared with the energy for diffusion on the terrace, in which a barrier ED of 0.503 eV had to be overcome in jumps. On Pd(110), interlayer movement of an adatom along the ½1
10 direction needed an energy EB of 0.42 eV for jumps, and 0.658 eV for exchange. To get over the step along the [001] direction on the same plane, a palladium atom needed an energy EB of 0.779 eV for jumps and almost the same, 0.777 eV, for exchange. Movement on the terrace was estimated as 0.366 eV for the ½1
10 direction and 0.776 eV

450

Diffusion in special environments

for cross-channel jumps in the [001] direction. On the Pd(111) plane, jumping over steps was more favorable than exchange, EB = 0.306 eV compared to 0.478 eV, but movement on the terrace had a much lower activation energy of 0.104 eV. Li and DePristo [42,43] utilized the corrected effective medium theory, CEM, to investigate surface diffusion on Pd(111) and compared this to atom incorporation into lattice steps. The barriers obtained for diffusion over B-type step edges, listed in Table 6.2, amounted to EB = 0.03 eV for a descending step, and 0.04 eV at a kink. What is interesting is that for palladium, the barrier to interlayer movement is lower than for diffusion on the flat surface. They also looked at small clusters of seven atoms or less and found the barrier to incorporation to be very high. However, the barrier decreased by an order of magnitude once the cluster size exceeded ten atoms and then started to be almost size independent. Although there is a bit of doubt about these findings, they do show that for palladium, as well as for nickel and platinum, the incorporation barrier is smaller for nineteen atom clusters than at kinks in the steps.

6.2.10

Silver(111): experiment Descending motion of adatoms from silver (111) was investigated in 1991 with STM by Vrijmoeth et al. [73] and from observations of the growth under different conditions, they were able to arrive at an additional step-edge barrier Ea of 0.150 ± 0.030 eV. They also looked at the effect of Sb on the step-edge barrier, and at a coverage of 0.08 ML found that the additional barrier for descending motion decreased by 0.03 eV. Smilauer and Harris [35] in 1995 came up with a quantitative relation for the onset of nucleation on islands, and arrived at values of Ea=0.13 eV to 0.23 eV for the additional step-edge barrier of Ag on Ag(111). At the same time, Meyer et al. [74] derived a different model to þ0:040 eV for the same material. analyze growth and found a step-edge barrier Ea of 0:150
0:020 Using the rate of nucleation on islands measured in STM experiments, Bromann et al. [75], the same year, found an effect of strain on the step-edge barrier. Based on the temperature dependence of the step-down diffusion of Ag adatoms deposited on different substrates they arrived at the results in Fig. 6.26; an additional step-edge barrier of 0.120 ± 0.015 eV for Ag on Ag(111), and 0.030 ± 0.005 for Ag on Ag islands on Pt(111). The difference between the step-edge barriers can be associated with differences

Ag/Ag island/Ag(111)

Ea = 0.12 eV 0.1 0

ED

(b) Energy (eV)

Energy (eV)

(a)

Ag/Ag island/Pt(111) Ea = 0.03 eV 0.17

0

Fig. 6.26

Potential energy of Ag adatom diffusion on (111) plane [75]. (a) Ag deposited on Ag(111). (b) Ag deposited on Ag island on Pt(111).

6.2 To descending step edges

451

in binding of Ag atoms on Ag(111) and on a monolayer of silver, or with strain relief close to the island edges. The authors suggested that strain relief rather than electronic coupling was responsible for the difference observed in the step-edge barrier. Markov [45] in 1997 derived an expression for the critical temperature for the transition from crystal growth by step flow to two-dimensional nucleation. Using experimental data from diffraction studies he was able to give the additional step-edge barriers Ea for Ag/ Ag(111); he obtained a value Ea=0.15 eV based on the experimental findings of Elliot et al. [76], Vrijmoeth et al. [73], and Rosenfeld et al. [77]. Based on findings of Brune et al. [78] and Broman et al. [75], he arrived at a value of Ea=0.12 eV for the additional step-edge barrier. A different approach was followed by Roos et al. [79] who studied the dependence of interlayer diffusion on the ratio of the diffusivity to the flux, D/Ff. For Ag on Ag(111), the step-edge barrier Ea was estimated at 0.040 eV, rather lower than previous values. In the year 2000, Roos and Tringides [80] investigated self-diffusion on the (111) plane of silver. From their RHEED measurements, which will not be described here, and Monte Carlo simulations, they concluded that the frequency factor for interlayer diffusion was much larger than for diffusion over the surface, by something like a factor of 100. This conclusion has been challenged a number of times, by Krug [81], by Morgenstern and Besenbacher [82], as well as by Heinrichs and Maass [83]. We forgo any further comments here but note that subsequent additional work by Chvoj et al. [84] called for further studies. Interesting STM experiments were carried out by Morgenstern et al. [85], who compared the filling of a vacancy island surrounded by an ascending step with the decay rate of an adatom island, sketched in Fig. 6.27. They found an additional stepedge barrier Ea of 0.13 ± 0.04 eV, with νs/ν0 = 10–0.6 ± 0.5, where ν0 is the frequency factor for atomic diffusion and νs for motion across the step. The measurements directly showed the energy required for descending motion, but the value obtained was an average over all types of steps, as well as for descent at kinks and corners. Recently Li et al. [86] looked at “wedding cakes” morphologies with STM. To account for such morphologies created in the Ag/Ag(111) system with KMC simulations they needed to introduce a non-uniform step-edge barrier. For a step of type B the value of the

Fig. 6.27

Schematics of geometry for measuring step-edge barrier [85]. (a) Vacancy island. (b) Adatom island, both surrounded by ascending steps.

Diffusion in special environments

In [lsland Density (cm–2)]

452

Fig. 6.28

26 24 22 20 18

0.0

0.3

0.1 0.2 Coverage Θ (ML)

Density of silver islands on Ag(111) as a function of the amount of deposited antimony (after van der Vegt et al. [87]).

500

Energy (meV)

400 ΔED 300 ΔEB 200 100 0

Fig. 6.29

Ea

0.00 0.05

0.10 0.15 0.20 Sb coverage (ML)

0.25

0.30

Changes in diffusion barriers of Ag on Ag(111) due to the presence of antimony. Ea gives additional step-edge barrier, ΔED is the additional barrier to surface diffusion. ΔEB is the change in the total step-edge barrier (after van der Vegt et al. [87]).

additional step-edge barrier Ea was 0.08 eV and for step A, Eawas twice bigger to explain experimental observations at 150 and 180 K. Van der Vegt et al. [87] examined the effect of antimony, amounting to ≤ 0.2 monolayers, on the homoepitaxial growth of Ag(111). The saturation density of islands, nx, at a

i=ðiþ2Þ F given coverage is found according to nucleation theory as in Eq. (2.15): nx / Df , where Ff is the rate of deposition and i the critical size of the nucleus. The deposition rate was fixed and, assuming that the critical nuclear size does not change, an increase in the island density must be due to a change in the diffusivity D. Antimony increased the silver island density exponentially, as in Fig. 6.28, and this was interpreted to be caused by an increase in the activation energy for diffusion, given by ED ¼ Eo þ 1:7Y;

(6:3)

where Θ is the amount of antimony deposited on the surface. Also estimated was the step-edge barrier, and the effect on it of the antimony on the surface. The changes in these quantities are shown in Fig. 6.29. The total energy barrier to escape, EB, increases. But it

6.2 To descending step edges

453

is given by the sum of the activation energy for diffusion ED plus the additional step-edge barrier Ea, and the diffusion barrier increases so rapidly with antimony concentration that the additional step-edge barrier actually diminishes.

6.2.11

Silver(111): theory Stoltze [41] used effective medium theory in 1994 for investigations of the Ag(111) plane and found jumping over steps easier than exchange, EB = 0.359 eVagainst 0.455 eV. Both values are higher than for movement on the plane, which proceeded with an activation energy ED of 0.064 eV. No information about the type of step was given. Li and DePristo [42,43] calculated barriers with the corrected effective medium theory, CEM; for crossing B-type steps by exchange they obtained the barriers listed in Table 6.2. All barriers for crossing the steps are higher than for diffusion over the flat terrace. For descending a perfectly straight A-type step by hopping an energy EB of 0.44 eV was required, while exchange on the same step was slightly lower at 0.31 eV. Descending a perfect type B step by hopping cost EB = 0.43 eV; however, exchange required only 0.06 eV. The energies were slightly lower on a 19 atom hexagon; hopping cost EB = 0.39 eVover an A-type step and 0.40 eV for a B-type, exchange was more likely on an A-type step with a barrier of 0.22 eV but only 0.02 eV on a B-type step. Descent on a zigzag step cost EB = 0.12 eV by exchange. Descending next to a kink on an A-type step required EK = 0.19 eV, while at a step of type B it was only 0.05 eV. These data are a bit uncertain since there is not perfect agreement between the two papers published by Li and DePristo. In 2002 Heinrichs and Maass [88] looked at the influence of interactions on epitaxial growth and found that the height deduced for the step-edge barrier was very sensitive to interactions. Based on the experimental findings of Brune et al. [78] for Ag on 1ML of silver deposited on Pt(111) they found that the additional step-edge barrier Ea can change from 0.068 ± 0.001 eV without interactions to 0.074 ± 0.002 eV when ring barriers between two adatoms are allowed. This paper showed clearly that adatom interactions need to be investigated in detail to arrive at proper values for step-edge barriers. Hong et al. [89,90] studied the effect of strain on the step-edge barrier on Ag(111) using the EAM potentials of Cai and Ye [91]. Their data are summarized in Fig. 6.30a. They found that strain had the opposite influence on the step-edge barrier compared to movement on the terrace. As already shown in Section 3.2.4, the surface diffusion barrier ED increases with lattice stress, but the step-edge barrier Ea monotonically decreased. Tensile stress makes descent easier while compressive stress makes it harder, probably due to the open structure of the steps. Haftel and Rosen [92] in 1998 looked at the descending motion of an Au adatom over steps of the Ag(111) surface with molecular dynamics and EAM potentials. The Au adatom moved over the terrace by hopping, with an energy ED of 0.102 eV. Haftel and Rosen were unable to detect any adatom hopping over A-type steps, hopping over B-type steps required an energy EB = 0.716 eV. Exchange over steps of both A- and B-type happened over a very low barrier; 0.051–0.061 eV for steps of type A and 0.047 eV for steps of type B.

0.26 0.04 0.25 (a)

0.24

0.03 0.98 0.99 1.00 1.01 1.02 1.03 1.04 Relative Lattice Constant

Fig. 6.30

0.05

Surface Diffusion Barrier (eV)

A step B step Adatom

0.27

Activation barrier (eV)

Diffusion in special environments

Step-edge barrier Ea(eV)

454

0.35

(b)

0.30 0.25 0.20

Surface diffusion barrier Step-edge barrier Ea

0.15 0.97 0.98 0.99 1.00 1.01 1.02 1.03 Relative Lattice Constant

Dependence of Ehrlich–Schwoebel barrier on lattice strain calculated with EAM for (a) two types of steps on Ag(111), (b) step on Ag(100) (after Hong et al. [89]).

Experimental values for descending motion from the Ag(111) surface seem to be in quite good agreement, unfortunately the mechanism of descent has not been uncovered. Theoretical values are confusing – one prefers hopping, the other exchange, and barriers are also not in agreement.

6.2.12

Silver(100) Studies of descending motion from Ag(100) started with the work of Stoltze [41] using effective medium theory. The jump down the probably step cost EB = 0.45 eV, and exchange required an even higher energy of 0.559 eV. Terrace movement required only 0.365 eV. In 1997 Merikoski et al. [44] looked at step characteristics on fcc(100) planes using EMT. The step-edge barriers EB were 0.481 eV compared with 0.516 for exchange, and higher than for movement on a perfect (100) plane, which cost 0.367 eV; these values are in quite good agreement with previous findings by the same method. For descending at a kink by jumps they found a value EK = 0.401 eV, which is comparable with 0.404 eV derived for exchange. Descending into kinks has lower energy requirements than descending at a straight step, but the barrier for interlayer motion was higher than the barrier for intralayer transport. The same year Yu and Scheffler [93] did densityfunctional theory calculations using both the local-density and the generalized gradient approximations. For an atom descending a step by jumps, they got an activation energy Ea = 0.18 eV higher than for diffusion on the terrace using LDA and Ea = 0.1 eV from GGA. But more interesting were values for exchange, which turned out to be comparable with terrace diffusion on this plane, 0.52 eV from LDA and 0.45 eV from GGA. Yu and Scheffler claimed that the lack of a barrier at the step explained the smooth 2D growth on the silver (100) plane. Kürpick and Rahman [94] using the embedded atom method with FBD and VC potentials calculated diffusion on the Ag(100) plane. They considered two kinds of steps for this plane, and . For the step using FBD potentials, hopping required EB = 0.59 eV against 0.64 eV for exchange, while VC potentials showed the opposite tendency: hopping needed EB = 0.70 eV and exchange 0.51 eV, so it is difficult to draw conclusions about the mechanism of descent based on these results. Even more

6.2 To descending step edges

455

interesting were the steps. Here using both potentials the energetically lower one was the exchange mechanism. From FBD, hopping had to overcome a barrier EB of 0.51 eV and exchange 0.38 eV; from VC, hopping needed 0.55 eV versus 0.31 eV for exchange. But more surprising is that the exchange over the edge required less energy than diffusion on the terrace, which faced a barrier ED of 0.48 eV, suggesting a negative barrier at the step edge, seen previously by Trushin et al. [57] for descent by exchange on a type step of Cu(100). This negative barrier would explain the rare existence of this type of step on the fcc(100) surface. Montalenti and Ferrando [69] calculated step-edge barriers with RGL potentials and for the descent of an Ag adatom on Ag(100) they found a barrier EB of 0.59 eV for jumps; for descent by exchange, most likely over a type edge, the barrier was very close, 0.55 eV. Both values are higher than the barrier to diffusion on the terrace, ED, which according to their estimate amounted to 0.43 eV. Hong et al. [89,90] studied the effect of strain on the step-edge barrier on Ag(100) using EAM potentials; their data are summarized in Fig. 6.30b. Similarly to Ag(111), they found that strain had the opposite influence on the step-edge barrier compared to movement on the terrace. From an analysis of Ag film roughness observed with a STM at 230 K, Stoldt et. al. [95] deduced the additional step edge barrier of 0.03 eV. Two years later the same group [96] again arrived at a much higher value of 0.07 eV from film roughness and the mean mound diameter. The situation of descending motion on Ag(100) is not certain, since half of the theoretical studies favor exchange, the other half jumping, and there is no experimental data available for verification. One thing seems to be clear – steps will not exist on the surface at temperatures where diffusion is not frozen, since as soon as an adatom reaches them incorporation will be immediate.

6.2.13

Silver(110) Just as for the silver(100) surface, there are no experimental data for descending motion from the Ag(110) surface. There are, however, a few theoretical results. These started with the work of Stoltze [41], who using EMT investigated two kinds of steps parallel and perpendicular to the channels of the anisotropic Ag(110) plane. Jumping down along the ½1
10 direction required an energy EB of 0.36 eV, while over steps the barrier EB was 0.646 eV. The anisotropic movement on the flat plane required 0.291 eV for the ½1
10 direction and 0.639 eV in the [001]. Exchange over steps was more demanding, EB = 0.537 eV, while for the [001] direction it was less favorable – 0.605 eV. Work was done also by Hontinfinde et al. [97] using RGL interactions that generally give reasonable results. For in-channel terrace motion their diffusion barrier ED was 0.28 eV, the activation energy for cross-channel motion by atom exchange amounted to 0.38 eV. This is to be compared with a barrier of EB = 0.56 eV for incorporation at an A-type step of {100} facets ( steps); incorporation proceeded by atom exchange since jumps required a much higher activation energy of 0.8 eV. For B-type edges of {111} facets ( steps), incorporation occurred by hopping over a barrier EB of

456

Diffusion in special environments

0.35 eV. The barrier is comparable with the energy of terrace exchange and only 0.07 eV higher than the activation energy for in-channel hopping. The barrier to descent over a step of type B increased strongly, to EB = 0.53 eV, if two atoms formed a dimer. Descending by exchange at a kink site at an A-type step required a barrier EK for incorporation of 0.38 eV on a large island and 0.29 eV on a small one. These estimates clearly indicate that kinks favor incorporation. Density-functional calculations for silver atoms diffusing over ascending lattice steps have also been done by Zhu et al. [33]. For in-channel motion, already illustrated in Fig. 6.21, the silver atom had to overcome a barrier of 0.68 eV to go up by exchanging with a lattice atom; the barrier downward was 0.54 eV. For cross-channel ascent by exchange the barrier was 0.62 eV and again 0.62 eV going down. Descending motion from Ag(110) seems to preferably proceed by jumping down along channels of the plane. Existing theoretical estimates seem to be in agreement that such movement would require ~0.07 eV, while cross-channel descending motion required a four times higher energy.

6.2.14

Tungsten Tungsten was the first material on which step crossing was investigated and the additional step-edge barrier was discovered. The reason is clear – tungsten is a very suitable material for FIM, and FIM has been the most useful technique for direct observations of single atom movement. The early findings have already been described in which a small additional barrier Ea~ 0.065 eV was discovered for self-diffusion on W(110) [16] and ~ 0.05 for W(321) [19]. For hetero-atom descent on W(110), a reflective barrier was also shown to exist but to be very small [21,22]. Here we will continue with later results. In 2002, Fu et al. [98], using the FIM, determined the rate of escape of palladium atoms from the (111) plane of tungsten at different temperatures, and arrived at an escape barrier of 1.63 ± 0.06 eV and a prefactor of 3.9 × 1013 ± 0.7 sec−1. Assuming a prefactor for diffusion of 10–3 cm2/sec, they came up with a diffusion barrier of 1.02 ± 0.06 eV for movement on the terrace, yielding an additional step-edge barrier of 0.61 eV. This large value should not be surprising, as the bcc(111) plane is bounded by quite rough steps. In the same year, Fu et al. [98,99] continued to look at the escape of palladium atoms from the W(110), the W(111), and from the W(211) plane, as shown in Fig. 6.31. For escape from W(110) they give an extra escape barrier of 0.27 ± 0.02 eV and a frequency prefactor of 6.3 × 1012 sec−1. For W(211), the additional activation energy for escape was given as 0.29 ± 0.08 eV, always assuming a prefactor of 3.7 × 1012 sec−1. These results should not be taken too seriously, as the fit to the data points in Fig. 6.31 for these characteristics is surprisingly poor. Additionally it was shown by Fink and Ehrlich [24] that on W(211) the barrier for W, Re and Si was not localized at the step. Such a possibility should also be taken into account in an analysis of the data for Pd. There are no theoretical efforts for deriving the step-edge barrier on tungsten surfaces.

457

6.2 To descending step edges

Pd/W(211) Ea = 0.29 ± 0.03 eV ν0B* = 3.7 × 1012sec–1

ln[Pb /(1–Pb)]

2.5 Pd/W(111) Ea = 0.61 ± 0.01 eV ν0B = 3.9 × 1013.0±0.7sec–1

0.5

–1.5

–3.5

–5.5 1.0

Pd/W(110) Ea = 0.27 ± 0.02 eV ν0B* = 6.3 × 1012sec–1 1.5

2.0

2.5

3.0 3.5 1000/T (1/K)

4.0

4.5

5.0

5.5

Fig. 6.31

Arrhenius plots for rate of descent of Pd over tungsten steps on W(111), W(110), and W(211). Listed are the additional step-edge barriers, and assumed prefactors indicated by * (after Fu et al. [99]).

6.2.15

Rhenium Not much information is available about atom behavior on rhenium, but an interesting FIM study was reported by Goldstein [100] in 1999. He was primarily concerned with surface diffusion on Re(0001), the only hcp metal studied, and in his work determined an activation energy for self-diffusion ED of 0.481 ± 0.019 eV and a prefactor of 6.13(× 2.6±1) × 10–6 cm2/sec. For a tungsten adatom on the same surface, diffusion occurred over a similar barrier of 0.478 ± 0.005 eV, with a more normal prefactor of 2.17(× 2.7±1) × 10–3 cm2/sec. More interesting in the present context, Goldstein found that at temperatures below 250 K, rhenium atoms diffusing over the surface were trapped on reaching the edge. For tungsten atoms, trapping occurred below 210 K. When the temperature was raised above these limits, the adatom detached from the edge and moved back to the center, an event observed many times. Under these conditions, collisions with the step edge were frequent, but the atom was never lost from the plane, suggesting the potential energy curve shown in Fig. 6.32. The plane is bounded by a substantial barrier EB, but adjacent to it is a trap of significant depth bounded on the other side by a lower barrier ER that obstructs the return to the center. Goldstein assumed that the prefactor for diffusion was constant over the entire plane up to the step edge, and in this way arrived at a return barrier ER of 0.57 ± 0.02 eV for Re and 0.58 ± 0.02 eV for W atoms, almost 0.1 eV higher than for diffusion in the central region. The step-edge barriers EB arrived at under the same assumptions were given as 0.68 ± 0.02 eV for Re and 0.83 ± 0.02 eV for W atoms. The qualitative behavior of atoms at the step edge of rhenium therefore somewhat resembles what has been found for the edge of the Pt(111) plane, and it is clear that direct examination of the behavior of single adatoms provides the most valuable information. Here it should be noted that Goldstein also roughly tested the mechanism by which an atom crosses the step edge. He deposited a tungsten atom on the surface, and then identified by field evaporation the atom attached

458

Diffusion in special environments

Ea

ED EB

Fig. 6.32

ER

Schematic energy of atom diffusing near edge of Re(0001) plane, deduced from observations of single adatoms [100].

to the edge after incorporation. In his only two trials, tungsten was found at the edge. This suggested that a simple jump had occurred at least on one kind of the steps bounding this plane, but more experiments are needed to demonstrate this unequivocally. Unfortunately, no reasonable attempts have been made at calculating the step-edge barriers.

6.2.16

Iridium: experiment Experimental investigations by Wang and Ehrlich [25,26] of W, Re, and Ir atoms overcoming A- and B-type steps on Ir(111) for the first time demonstrated descent by exchange; these were described earlier, in Section 6.2. It is clear from this study that descent from Ir(111) is associated with a reflective barrier, since higher temperatures are required to initiate descent. Unfortunately, Wang did not carry out investigations of the energy for this process. Here we will continue our survey with subsequent findings for iridium surfaces. In 1998, Fu et al. [101] examined atom escape for three systems: rhodium and iridium atoms from Ir(100), and iridium atoms from the iridium (111) plane. FIM measurements were done at a number of different temperatures, yielding Arrhenius plots from which detailed kinetics were derived, as shown in Fig. 6.33. For rhodium leaving the Ir(100) surface, the activation energy EB was 0.84 ± 0.08 eV, with a frequency prefactor of 2.4 × 1013 ± 1.4 sec−1. The diffusion characteristics of rhodium on the same surface were quite close to these values, ED=0.80 ± 0.08 eV for the diffusion barrier and a prefactor of 2.09 × 10–3 ± 1.5 cm2/sec, which makes it quite uncertain that for this system there really is a step-edge barrier Ea. For iridium atoms on the same plane, the barrier for escape EB was 0.72 ± 0.08 eV, compared with an activation energy for diffusion ED of 0.74 ± 0.02 eV; the prefactors were given as 5.3 × 1013 ± 1.6 sec−1and 7.30 × 10–3 ± 0.5 cm2/sec for diffusion. Here again there is doubt about the existence of a step-edge barrier. For iridium leaving Ir(111), Fu et al. found a barrier EB of 0.41 ± 0.03 eVand a frequency prefactor of 1.8 × 1013 ± 1.1 sec−1. To arrive at a step-edge barrier Ea, they picked a value of 0.21 ± 0.03 eV for the diffusion barrier given by Kellogg [102], but this is clearly too low. The self-diffusion characteristics on Ir(111) are an activation energy of 0.289 ± 0.003 eV, and a prefactor Do = 3.5(x1.4±1) × 10–4 cm2/sec [103], so that the

459

6.2 To descending step edges

Ir/Ir(001) Ir/Ir(111) EB = 0.72 ± 0.08 eV EB = 0.41 ± 0.03 eV ν0B = 5.2 × 1013±1.6sec–1 ν = 1.6 × 1013±1.1sec–1 0B

1

ln[Pb /(1–Pb)]

0

Crossing the step

–1

–2

or

–3

–4 3.0

Rh/Ir(001) EB = 0.84 ± 0.08 eV ν0B = 2.4 × 1013±1.4sec–1 3.5

4.0

4.5

5.0 5.5 6.0 1000/T (K–1)

6.5

7.0

7.5

Fig. 6.33

Arrhenius plots for the descent of atoms from Ir(100) and Ir(111) planes (after Fu et al. [101]).

Fig. 6.34

Sites at which Ir atoms were located on Ir(111) cluster of ~ 150 atoms. Open circles indicate location after deposition at ~ 30 K, black circles after diffusion (after Fu et al. [101]).

additional step-edge barrier Ea amounted to 0.12 eV. Here it should be noted that the (111) plane of iridium is similar to that of platinum: both have a central region populated by adatoms but surrounded by an empty zone. Sites at the very edges of the plane bind atoms, as shown in Fig. 6.34, but no estimates were made of how this affected the escape process [101,104].

460

Diffusion in special environments

The mechanism of descent was not examined, but based on the exchange of Ir on Ir(100), the authors suggested that descent would be by exchange as well. For Rh on Ir(100) they mentioned that jumps might be possible, but this idea was not investigated separately. To determine the difference in the barrier for an Ir atom to descend over steps of type A and B on Ir(111), the authors looked at 58 descending runs in which atoms crossed step A 18 times and step B 40 times. Based on this observation the difference in activation energy was estimated as 0.012 eV. The statistics are rather poor, and the mechanism of descent was not probed, but the results are in agreement with the earlier work of Wang and Ehrlich [25]. A number of theoretical studies suggest that vacancies, corners, and kink sites accelerate the descent of atoms. This proposition has been studied directly by Wang [105], who recently tested the effect of edge imperfections on the rate of incorporation. In the first experiments, an Ir atom was deposited on an Ir19 cluster sitting on an Ir(111) plane. The diffusion path of the atom at 160 K was observed, and as shown in Fig. 6.35, incorporation at an A-type step occurred after ten 5 sec heating periods. The experiment was repeated 28 times, and 13 times the atom incorporated at a type A step, showing equal probability of incorporation to step A and B for this cluster. Here it is worth mentioning that the FIM should be able to detect changes in the incorporation barrier bigger than 5%. Next an atom was attached to the side of an Ir19 cluster formed on Ir(111), shown in Fig. 6.36, to introduce a kink. The atom in the lower left-hand side should play the role of catalyst if incorporation is preferred at this defect. An adatom was deposited on the surface, and its path in diffusion after heating to 160 K is shown in Fig. 6.36, but even though the adatom twice was close to the “kinked edge,” incorporation eventually

Fig. 6.35

Incorporation of Ir atom deposited on top of Ir19 cluster, showing the track of the atom in diffusing at 160 K [105]. Incorporation finally takes place at lower left step edge.

6.2 To descending step edges

461

Fig. 6.36

Ir atom incorporation into Ir20 cluster, with one atom attached to central Ir19 core [105]. (a) FIM image of Ir20 cluster. (b) Shows atom after deposition. (c)–(g) Atom movement around edge after heating to 160 K. (h) Atom has incorporated into straight edge after 13 heating intervals.

Fig. 6.37

Ir atom incorporation into Ir18 cluster [105]. (a) FIM image of Ir18 cluster. Ir atom has been placed on top of cluster in (b), and was then heated for 5 sec at 120 K, moving atom close to edge in (c). (d)–(g) Motion along edge after heating to 160 K. (h) Atom incorporates into straight edge after nine heating intervals.

occurred into a straight edge opposite to the defect. Out of a total of 23 repetitions of this experiment, the adatom on the cluster incorporated 21 times into straight edges, 8 of them of type A. If the defect had any effect on atom behavior it proved to be small, with less than a one percent reduction of the incorporation barrier. It appears that the atom added to the straight step of an Ir19 cluster did not provide an especially low-energy path for incorporation. Also examined was the activity of different types of kink sites which calculations have shown to be very active in promoting atom attachment into a step. The kink was created by removing an edge atom from an Ir19 cluster. Measurements of atom diffusion on an Ir18 cluster, again on Ir(111), are shown in Fig. 6.37. The atom again migrated around on the cluster, passing near the kink site, but incorporation once again occurred into a straight edge rather than into the kink. This type of incorporation was repeated 27 times. In these trials, only one ended with the atom incorporating at the defect site,

462

Diffusion in special environments

Fig. 6.38

Iridium atom incorporation into Ir55 cluster, showing the track of the atom diffusing over the cluster [105]. Incorporation takes place at upper right edge.

suggesting a rather higher barrier to incorporation there. Finally an Ir55 cluster was created by evaporating all 6 corner atoms from a cluster of 61 Ir atoms. On this cluster the same procedure was followed, as indicated in Fig. 6.38. In 51 attempts, atoms incorporated in the empty corner only 4 times. This observation suggested a barrier at the kink rather higher than at a straight edge. The last test was done on a cluster of 63 atoms created by adding two atoms to Ir61 to create a kink. In 29 repetitions, atoms incorporated at kink sites three times, once in a kink created at a B-type edge and twice into a kink on an A-type edge. Here kinks might be slightly more favorable than at ordinary edges. Some other trials of incorporation were also done, but overall there was no indication of defects in the cluster edge significantly promoting incorporation. This, of course, raises serious questions of how to understand the disagreement with calculations showing preferred incorporation into defect sites, which may again have something to do with the different behavior of straight steps and steps at small islands, but this is clearly a matter that will have to be examined.

6.2.17

Iridium: theory RGL interactions were used by Trushin et al. [106] in estimates of self-diffusion on Ir(111). On a terrace the activation energy ED was 0.20 eV, but to go over the step and

463

6.2 To descending step edges

(b) –2

–3

–3

–4

Energy E(eV)

Energy E(eV)

(a) –2

L-J

–5

–4

–5 RGL

RGL –6 –7 –8 Fig. 6.39

L-J

–6

–6

–4

–2

0 2 y(Å)

4

6

8

–7 –8

–6

–4

–2

0 2 y (Å)

4

6

8

Diffusion of Ir atom on Ir(111) plane. Top: diffusion path over step. Bottom: energy along diffusion path. (a) A-type steps. (b) B-type steps. Dashed line: Lennard-Jones calculations. Solid line: RGL estimates (after Trushin et al. [106]).

incorporate required more energy, as is apparent in Fig. 6.39. For A-type edges, made up of (100) facets, atoms had to jump over a barrier EB of 0.97 eV to incorporate; exchange required a much higher energy of 1.75 eV. Incorporation at a kink site occurred somewhat more easily, over a barrier EK of 0.83 eV for atom jumps and 1.50 eV for atom exchange, as shown in Fig. 6.40. The situation at B-type edges, of (111) facets, is different. Jumps over the step required an energy EB of 0.98 eV, but exchange with a lattice atom took place more easily over a barrier of 0.74 eV. Incorporation at a kink site occurred over an energy EK of 0.83 eV for a jump, and 0.90 eV for atom exchange. Here we have to note that the estimate for the self-diffusion barrier ED made with RGL potentials is below the experimental value of 0.289 eV [103]. Studies of iridium atoms incorporating into a step edge on Ir(111) have also been done by Kürpick [107], who used RGL potentials to explore atom behavior at straight edges as well as at steps with vacancies. The expected behavior at both A- and B-type steps is

464

Diffusion in special environments

(b)

–3

–3

–4

–4

Energy E(eV)

Energy E(eV)

(a)

–5 RGL step –6

–5 RGL step RGL kink

–6

RGL kink

L-J kink

L-J kink –7 –8

–6

–4

–2

0 2 y (Å)

4

6

8

–7 –8

–6

–4

–2

0 2 y(Å)

4

6

8

Fig. 6.40

Diffusion of Ir atom on Ir(111) with descent over kinked step. Top: diffusion path over step. Bottom: energy along diffusion path. (a) A-type step. (b) B-type step (after Trushin et al. [106]).

Fig. 6.41

Ir atom descent from Ir(111) surface [107]. (a) Initial position of adatoms at A- and B-type steps. (b) Final position after atom hop. (c) Final position after atom exchange.

shown in Fig. 6.41. However, in molecular dynamics studies she did not obtain any incorporation events at straight steps at 900 K. Using transition-state theory she found the energy involved in such transitions was quite high, as is apparent in Table 6.3. At (100) or A-type steps, jumps to incorporation required an activation energy EB of 0.90 eV and a prefactor of (7.4 ± 0.1) × 10–4 cm2/sec, whereas exchange occurred over a much higher

465

6.2 To descending step edges

Table 6.3 Kinetic parameters for descent of Ir atom at Ir(111) step edges [107]. (100) facet (A-type): Straight step Jump

Ex

Kink Jump

Ex

Vacancy Jump

Ex

EB (eV) 0.90 1.58 0.77 0.85 0.75 (7.5 ± 0.1) × 10−4 DoB(cm2/sec) (7.4 ± 0.1) × 10–4 (2.7 ± 0.1) × 10–3 (111) facet (B-type): Straight step Jump

Ex

Kink Jump

0.91 1.03 0.77 EB (eV) DoB(cm2/sec) (7.0 ± 0.1) × 10−4 (5.6 ± 0.1) × 10−3

0.82 (2.1 ± 0.1) × 10−3

Vacancy

Ex

Jump

Ex

1.45

0.75 (7.4 ± 0.1) × 10−4

1.37 (2.2 ± 0.1) × 10−3

barrier of 1.58 eVand a prefactor of (2.7 ± 0.1) × 10–3 cm2/sec. At (111) facets, B-type steps, jumps occurred over a barrier of 0.91 eV with a prefactor of (7.0 ± 0.1) × 10–4 cm2/sec, while exchange involved an activation energy of 1.03 eV and a prefactor of (5.6 ± 0.1) × 10–3 cm2/sec. These values are to be compared with a barrier of 0.24 eV to diffusion calculated for the (111) plane. The situation was different when vacancies at the step were present, as suggested in Fig. 6.42. At an A-type step, descent by an atom jumping at a vacancy occurred over a barrier EV of 0.75 eV, with a prefactor of (7.5 ± 0.1) × 10–4 cm2/sec. Exchange involved a higher barrier of 0.82 eV and a prefactor of (2.1 ± 0.1) × 10–3 cm2/sec. Descent at a B-type step again occurred over a barrier EV of 0.75 eV, with a prefactor of (7.4 ± 0.1) × 10–4 cm2/sec, compared to a barrier for exchange of 1.37 eV and a prefactor of (2.2 ± 0.1) × 10–3 cm2/sec. Descent at a step with several adjacent vacancies took place over a somewhat higher barrier than when only a single vacancy was present. For exchange, the prefactor was 3 to 8 times higher than for hopping and in molecular dynamics simulations exchange on a type A step with a single vacancy was observed at 1100 K. This was explained by the Arrhenius plots in Fig. 6.43 for hopping and exchange, which intersect at 788 K so that exchange might get favorable at higher temperatures. These results are interesting, as they differ from experiments on clusters, for which descent, presumably by exchange, occurred preferentially at B-type edges. This may indicate a difference in atom descent at straight steps and at steps of small islands. On Ir(111) steps direct observations of step-edge barriers have been made and the data are in good standing, but diffusion over iridium needs much more attention. Theory for movement on the flat surface predicts values comparable with experiment. For descending motion, however, the predicted additional step-edge barrier Ea is huge, around twice the value for surface movement. The existing reports also are not in agreement about the mechanism of descent.

466

Diffusion in special environments

Fig. 6.42

Ir atom descent from Ir(111) plane with edge vacancies [107] (a) and (b) for atom hopping, (c) and (d) for atom exchange transition. (e)–(h) Atom descent when edge atoms are missing.

–10

1000

333

HOPPING EXCHANGE

–15 InD (cm2/sec)

Temperature T(K) 667 500 400

–20 –25 –30 –35

Fig. 6.43

1.0

1.5 2.0 1000/T (K–1)

2.5

3.0

Arrhenius plot for descent from Ir(111) plane with {100} faceted step edge with one vacancy, for hopping (solid line) and exchange (dotted line). Intersection at 788 K (after Kürpick [107]).

6.2 To descending step edges

6.2.18

467

Platinum: experiment In 1995 Smilauer and Harris [35] came up with a quantitative relation for the onset of nucleation on islands, and arrived at values for the additional step-edge barrier Ea of 0.15 eV to 0.27 eV for Pt on Pt(111). At the same time, Meyer et al. [74] derived a different þ0:050 eV for Pt on Pt(111). model to analyze growth and found a step-edge barrier Ea 0:165
0:020 Markov [108] in the same year derived another relation for second layer nucleation in homoepitaxy and arrived at two values for the step-edge barrier on Pt(111) from the experiments of Bott et al. [109]: Ea = 0.31 eVand 0.44 eV. The lower value was attributed to A-type steps, (100) facetted, and the higher one to passage over B-type steps. The following year, Markov [45] looked at the transition from crystal growth by step flow to two-dimensional nucleation, and using experimental data from diffraction studies done by Poelsema et al. [110] he arrived at a step-edge barrier of Ea ≈ 0.21 eV. A detailed FIM probe of how atoms escape from a plane was presented by Kyuno and Ehrlich [111] for platinum atoms on Pt(111). The Pt(111) surface is like Ir(111) in that only the center is occupied by adatoms; this center is surrounded by an empty zone of higher energy, which extends almost to the boundary of the plane. Atoms can again be held at the plane edge, as illustrated in Fig. 6.44. Kyuno and Ehrlich measured the rate at which platinum atoms escaped from this plane at temperatures from 120 to 140 K. The results are shown in the Arrhenius plot in Fig. 6.45, which yields an activation energy of 0.32 ± 0.01 eV. To properly interpret this figure it is necessary to know something about the potential energy variations on going across the plane. A rough diagram of the potentials close to the edge is shown in Fig. 6.46. The rate of escape re can be written as re ¼ PE ν0B expð
EB2 =kTÞ;

(6:4)

where PE is the probability that an atom occupies an edge site, ν0B is the frequency factor, and EB2 is the barrier to escape and incorporate. From observations of atom transitions, Kyuno was able to arrive at a barrier EB2 (which prevents atoms from incorporating from sites next to the step) equal to 0.36 eV, and for the barrier EB1 in Fig. 6.46 which confines atoms to the interior a value of ~0.33 eV. The activation energy for diffusion ED at the center of the Pt(111) plane has been found as 0.26 eV, so that the barrier at the step edge

Fig. 6.44

Location of Pt on Pt(111) plane [111]. (a) Positions at which Pt5 cluster was observed, all located in inner region. (b) Single platinum atoms were found either in center or at edges.

468

Diffusion in special environments

1

T (K) 130

140

120

Escape rate 1/τ (sec–1)

1/τ0 = 7.3( × 2.2±1) × 1010sec–1 E ‡ = 0.32±0.01 eV 10–1

10–2

1/τ = 1/τ0 exp(–E ‡/kT ) 10–3 7.4 7.8 1000/T (K–1)

7.0

Fig. 6.45

8.2

8.6

Arrhenius plot for the rate of Pt atom escape from Pt(111) island with radius of ~ 20 nearestneighbor spacings [111].

Inner region

ED

Empty zone

Edge

EB 2

EB 1

ER Δ

Fig. 6.46

Schematic of energy barriers to Pt atom jumps on Pt(111) cluster, sketched below [111]. Dotted curve indicates values were not well known.

EB is 0.1 eV higher. Direct measurement of the rate of escape delivered an activation energy Ea of 0.06 ± 0.01 eV above that for diffusion. This appears to be one of the first more reliable determinations of the potential opposing escape. A year later, in 2000, Krug et al. [112] developed a new theory for nucleation on twodimensional islands based on the residence time of an adatom on that island. For CO decorated Pt(111) the step-edge barrier Ea, derived from work by Kalff et al. [113], was 0.36 eV, as indicated in Fig. 6.47, and dropped significantly, to 0.12 eV, as the contamination was removed. The value predicted from the procedure described by Tersoff et al.

469

6.2 To descending step edges

0.4

TDT

600

0.2

0.1

400

LAM

Critical island size

Step Edge Barrier Ea (eV)

800 0.3

200

0.0 10–11

10–10

10–9

pco Fig. 6.47

Step-edge barrier (full symbols) and critical island size in their dependence on CO. Full circles from formula of Krug [112], triangles from Tersoff et al.[114] (after Krug et al. [112]).

[114], labeled TDT, differed significantly in the higher effect of CO, 0.28 eV compared with 0.36 eV. With a decreasing influence of impurities, the two analyzing techniques started to give comparable values, 0.15 eV from TDT compared with 0.12 eV from the lonely atom method (LAM) of Krug et al. [112]. From an analysis of multilayer growth on Pt(111) in the presence of a partial CO pressure of 1.9 × 10–9 mbar at 440 K they found a comparable barrier Ea of 0.33 eV. According to Krug et al. [112], the lonely atom method can be trusted only in the presence of a strong step-edge barrier, which is the case for a high CO pressure; in this region Tersoff et al. underestimated the value of the stepedge barrier. Krug et al. also claimed that for small pressures the LAM method was unreliable, as it underestimated the nucleation rates. However, surprisingly enough, the value obtained in the limit of small pressures agreed reasonably with the barrier measured directly by FIM. Very interesting examinations of cobalt deposits on a Pt(111) surface, on which vacancy islands had been created, were done by Lundgren et al. [115] using atomic resolution STM. The image of platinum and cobalt atoms can be distinguished since Co appear to be darker than Pt. Lundgren et al. deposited 0.04 ML of cobalt and found cobalt atoms inside the vacancy islands at the edge as well as in the second row from the edge, shown in Fig. 6.48. The presence of atoms in the second row suggested exchange as the mechanism of step descent. To get more information about the mechanism, the amount of Co at the edge and in the second row from the edge was investigated as a function of the vacancy size, as in Fig. 6.49. The number of Co atoms at the edge agreed with that expected from the rate of deposition and this amount depended on the size of the vacancy island. The number of atoms situated behind the first row of platinum atoms did not depend on the size of the vacancy island, and was associated with the atoms deposited on top. However, the number of descending atoms did depend on the step length. Since the number of atoms embedded in the second row was independent of island size, Lundgren et al. concluded that these atoms had diffused over the top and had exchanged with kink or corner platinum atoms from the lattice. Additionally, Co atoms liked to assemble in rows,

470

Diffusion in special environments

Fig. 6.48

STM image of vacancy island on Pt(111) surface. (a) Island formed in sputtering. (b) 0.04 ML of Co deposited on Pt(111) surface. (c) As in (b), but a different part of the surface. White arrows indicate atoms at the edge (type 1), black arrows show atoms behind the edge (type 2) (after Lundgren et al. [115]).

which also supported descents over kinks, since such exchange creates shifts, shown in Fig. 6.50. The authors never observed Co atoms in the second row, next to long straight steps without kinks. The Co atoms were not found uniformly all around the vacancy cluster – they appeared to be close to the corners of the island. The conclusion was that Co did not jump over the step-edge barrier. Instead, exchange occurred and was

6.2 To descending step edges

471

Fig. 6.49

Plot of type-1 and type-2 Co atoms against the vacancy island size. Dashed line gives number of Co atoms calculated to be in the vacancy island (after Lundgren et al. [115]).

Fig. 6.50

Models for exchange processes at corners, and kink creation. (a) Creation of kink by exchange. (b) Kink arises from Pt atom in corner (after Lundgren et al. [115]).

472

Diffusion in special environments

accelerated at the corners, whereas at straight steps atom descent took place much more slowly with a mechanism not uncovered by this investigation.

6.2.19

Platinum: theory The study of platinum steps started with calculations by Villarba and Jónsson [116] who made extensive estimates, using the embedded atom method, of the energy barriers important in the incorporation of platinum atoms into steps on Pt(111). On the Pt(111) plane itself the activation energy for diffusion ED amounted to 0.08 eV. They found that hopping over a step is highly unlikely, due to the high energy barrier EB of 0.7–0.8 eV. Of special interest are the barriers for an atom approaching A- or B-type straight, long steps shown in Fig. 6.51. Close to either step edge there is a very considerable alteration of the potentials. Exchange descent at the straight step required an energy EB of 0.30 eV from an fcc site and 0.25 eV from an hcp site at the A-type edge; at a B-type edge the barrier from an hcp site was 0.18 eV. Descent at the A-type edge was more demanding, due to the bigger number of atoms influenced by such a movement. In the exchange of an adatom at (a) type A step

1

(b) 0.3

10 7–6 3–2 9–8 5–4 1

2–3 6–7 10 8–9 4–5

type B step

EXCHANGE from HCP edge site

Diffusion to step edge Diffusion atop flat terrace

ENERGY (eV)

0.2 type A step

0.1 0.0 10 0.2

9

8

7

6

5

4

3

2

type B step

EXCHANGE from HCP edge site

0.1

1 0.0

10

9

8

7

6

5

4

3

2

–0.1 0

2

4

6

8

10

12

14

16

DISPLACEMENT (Å) Fig. 6.51

Platinum atom behavior on Pt(111) surface. (a) Model of A- and B-type steps. (b) EAM calculated energy of platinum atom diffusing on flat (111) surface (dashed) and over steps (solid) by atom exchange (after Villarba and Jónsson [116]).

473

6.2 To descending step edges

Table 6.4 Barriers for exchange descent of Pt from compact Pt islands [116]. Cluster

Barrier EB (eV)

Trimer Heptamer

0.49 0.48 (fcc)

0.29 (hcp)

19-atoms 37-atoms 61-atoms 91-atoms 127-atoms 169-atoms Straight Edge Near Kink

A-type 0.38 0.38 0.33 0.32 0.31 0.31 0.30 0.08

B-type 0.03 0.06 0.14 0.16 0.17 0.17 0.18 0.06

an A-type edge, the adatom must go around an underlying atom, displacing the nearest and next-nearest edge atoms. For descent by atom exchange at kink sites, they estimated a much lower barrier EK of 0.08 eV at an edge of type A and 0.06 eV at a B-type edge. For a B-type kink the activation energy was lower than for movement on the terrace. They also estimated how the barrier to incorporation varied with cluster size. As is apparent from Table 6.4, for B-type edges the barrier EB increases from 0.03 eV for a 19 atoms cluster to 0.17 eV at a 169 atoms cluster. Over the same range of cluster sizes the barrier EB at A-type edges decreases somewhat, from a high of 0.38 eV to 0.31 eV. Villarba and Jónsson also compared incorporation at a straight B-type step with what happens on a 37 atoms cluster, as shown in Fig. 6.52. What is apparent here is that for an atom on the island that has reached the edge (site 2), the descent is easier than a return to the interior. For displacing a corner atom of a 19 atom island by exchange the barrier was only 0.19 eV. The barrier for an adatom descending from small clusters, such as trimers and heptamers, was higher than at the straight edges, 0.49 eV for descending from a trimer and 0.29 eV from an heptamer starting at an hcp site. Although the embedded atom method does not give good diffusion barriers on Pt(111), these estimates are certainly the most extensive and seem to show effects revealed in experiments. It should be noted that Siswanto and Jónsson [117] more recently have done density-functional calculations which show behavior quite similar to the embedded atom work, as seen in Fig. 6.53. At roughly the same time, Wang and Fichthorn [118] performed molecular dynamics simulations with the corrected effective medium theory (CEM) for incorporation of platinum atoms into a descending step on Pt(111). On a 3 × 3 cluster with A-type edges, incorporation always occurred by exchange with an energy EB of 0.03 eV, which is a slightly lower value than the ED = 0.038 eV calculated by the authors for movement on the clean relaxed surface. They also did simulations on 6 × 6 clusters, and again exchange was the significant mechanism. Incorporation by exchange was seen on both types of edges, but exchange at B-type steps was more frequent. On the larger 6 × 6 clusters a sizeable number of atoms did not incorporate, and they could not exclude the

474

Diffusion in special environments

0.3 DIFFUSION TO STEP EDGE DIFFUSION ATOP FLAT TERRACE

DESCENT from HCP edge site

0.2 0.1 ENERGY (eV)

1 0.0 10 0.2

9

8

7

5

4

3

6 5 4 3 37-atom island

2

0.1 6

1 DESCENT from HCP edge site

0.0

2

–0.1 0

2

4

6

8

10

12

14

16

DISPLACEMENT (Å) Fig. 6.52

Energy of Pt atom diffusing on Pt(111) surfaces. Comparison of behavior at straight B-type step and at cluster of 37 atoms. Dashed curve indicates diffusion on flat (111) plane (after Villarba and Jónsson [116]).

0.4 0.2 ENERGY (Å)

0.0 –0.2

a.

–0.4 1

b. 2

–0.6

1

–0.8

2

–1.0 –1.2 2

Fig. 6.53

4 6 DISTANCE (Å)

8

Density-functional calculations of Pt atom diffusing over B-type step on Pt(111) surface (after Jónsson [117]).

possibility of jumps occurring. From static potential energy estimates for atoms diffusing toward a step edge they found the energy at a site two steps from the edge was considerably higher than in the interior for both 7 × 7 and 10 × 10 clusters, as in Fig. 6.54. A year earlier Wang and Fichthorn [119] looked at descent by hopping from 3 × 3 and 4 × 4 islands; the barriers EB were much higher, 0.50 eV for a 3 × 3 island and

475

6.2 To descending step edges

–4.20 (b) –4.21

–4.20

Energy (eV)

Energy (eV)

–4.18 (a)

–4.22

–4.24

–4.22 –4.23 –4.24 –4.25

6 4 2 Distance from Edge (unit = 0.5 nearest-neighbor distance) 8

Fig. 6.54

6

4 Distance from Edge (unit = 0.5 nearest-neighbor distance)

2

Energy profile near edge for Pt atom on Pt(111). (a) At B-type edge of 7 × 7 (triangles) and 10 × 10 cluster. (b) At A-type edge of 7 × 7 island. Dashed curves indicate energy on a flat surface (after Wang and Fichthorn [118]).

0.51 eV for a 4 × 4. They found exchange to be a much more probable mechanism for descent, but also that exchange might be enhanced by momentum transfer between an impinging atom and the island. In the same year Stoltze [41] used effective medium theory to look at platinum steps. He calculated the energy required for moving over different kinds of steps. On Pt(100), an energy EB of 0.755 eV was needed to jump over the step, 0.066 eV higher than for jumps on the flat surface. Exchange over this step was slightly more demanding at EB = 0.797 eV. Jumps in the [001] direction on Pt(110) required EB = 0.944 eV, while exchange occurred over a barrier of 1.294 eV. Exchange along the ½1
10 direction cost a
bit less, EB = 0.782 eV. Movement on the plane in the ½110 direction proceeded over an energy barrier ED of 0.42 eV while in the [001] direction diffusion took place over a barrier of 0.945 eV; the latter value was the same as for jumping down to a lower level in the [001] direction. According to Stoltze, jumping over a step edge was more favorable on Pt(111) than exchange, EB = 0.363 eV compared with 0.470 eV for exchange. Both values were much higher than for movement on the terrace, which needed only ED = 0.159 eV. It should be noted, however, that this value was much smaller than what has been obtained in experiments. Monte Carlo estimates of the diffusion of platinum atoms on Pt(111) were made by Jacobsen et al. [120] in 1995 also using effective medium theory. Their estimates of the energy variations as an atom approaches and then crosses a lattice step are shown in Fig. 6.55. The change in potential two spaces away, as an adatom approaches a descending step, is clear from this figure. Incorporation proceeded by an atom jumping over a step of type A and by exchange over B-type steps. For the self-diffusion barrier ED on Pt(111) they estimated a value of 0.16 eV, which they recognized as too low. For motion over the edge at an A-type step they found a barrier EB of 0.41 eV, compared to an exchange barrier EB = 0.37 eV over a B-type step. The lowest activation energy for incorporation, of EK = 0.26 eV, was found at a kink site in a B-type step. Li and DePristo [42,43] using the corrected effective medium theory, CEM, obtained the barriers to self-diffusion on Pt(111) surfaces. Movement over B-type step edges are

476

Diffusion in special environments

Energy E(eV)

E

G

1.0 F 0.5 E 0.0 0.0

Fig. 6.55

F

G 5.0 10.0 Nearest-neighbor distances

Platinum atom diffusion over Pt(111) clusters. Top: Model for diffusive atomic motion over step. Bottom: EMT calculations of energy changes accompanying above movement (after Jacobsen et al. [120]).

listed in Table 6.2. For descent of platinum by exchange they obtained a value for overcoming a straight step (EB = 0.044 eV) and kink (EK = 0.012 eV) lower than for movement on the terrace. These findings have to be treated carefully since most of them do not agree with any other results. They also showed that for platinum the incorporation barrier was smaller for nineteen atom clusters than at kinks in the steps. Ab initio simulations of self-diffusion of platinum atoms over step edges on Pt(111) were made by Feibelman [121] using the Vienna simulation package (VASP) [122–124]. The atoms preferred sitting in fcc sites on the surface by 0.21 eV, rather larger than the >0.06 eV found in experiments [125]. Diffusion on a (111) terrace in this study involved overcoming a barrier ED of 0.29 eV. For jumping over the edges the energy EB required was 0.53 eVat an A-type step and 0.80 eVat a B-type step. Exchange down an A-type step involved overcoming a barrier EB of 0.31 eV, whereas down a B-type step the barrier proved to be higher, 0.64 eV. The step edge barrier Ea, according to Feibelman, was therefore 0.02 eV for an A-type step and interlayer movement proceeded preferentially via step A. The mechanism for descending A- and B-type edges is, of course, the reverse of what had previously been found in experiments for hetero-descent on Ir(111). Self-diffusion on small islands on Pt(111) has been the topic examined by Ma´ca et al. [71,72,126] using RGL interactions between atoms. For platinum on Pt(111) the diffusion barrier ED proved to be only 0.17 eV, much lower than the value of 0.26 eV from experiments. For jumps over the straight edges the activation energy EB was found to be similar for the two types: 0.53 eV for an A-type of step and 0.52 eV for B-type. For exchange at straight B-type steps the barrier EB was considerably smaller, 0.40 eV. The activation energy EB for exchange on A-type steps was also lower than for jumps: 0.50 eV. The presence of kinks seemed to influence interlayer transport. The energy EK for descent by exchange over type B steps with kinks was as low as 0.26 eV, but only 0.42 eV on type A. Ma´ca et al. also looked at descent from 3 × 3 and 6 × 6 islands on Pt(111). For small Pt islands no change was found in the barrier for jumping, but there were huge differences

6.2 To descending step edges

477

for exchange. In the case of 3 × 3 islands exchange required EB = 0.26 eV for type A steps and 0.13 eV for type B. The values for 36 atom islands were between those for 3 × 3 island and long straight edges: EB = 0.30 eV for A-type steps and 0.16 eV for B-type. This reduction as the size becomes smaller was stronger at B-type edges than at edges of type A. What is of interest here is that transitions down steps are likelier at B-type steps than at steps of type A, the opposite of what has been found by Feibelman [121]. It must be noted, however, that the results of Ma´ca et al. differ significantly from the experimental work of Kyuno [111], which yielded rather larger values. Leonardelli et al. [127] made calculations with EAM potentials for platinum diffusion over steps on Pt(111). They observed that in considering the transition state, Pt sank deeper into the surface, creating four fold metastable adsorption places which lowered the energy barrier for diffusion ED to 0.046 eV. The metastable four fold adsorption places were not observed in direct experiments by Gölzhäuser and Ehrlich [125] where two kinds of adsorption sites (fcc and hcp) were clearly distinguished. According to Leonardelli nickel atoms, however, deposited on the surface did not create such metastable sites and the activation energy ED was around 0.178 eV. For descent over A-type steps by a platinum adatom at an fcc site, the activation energy for exchange EB was 0.29 eV; it was 0.28 eV if the atom at first was in an hcp site. For movement over B-type edges the barriers were smaller, EB = 0.17 eV if starting from an fcc site and 0.19 eV from hcp. Leonardelli arrived at a step edge barrier of 0.08 eV for B-type edges and 0.20 eV for A-type steps. All these estimates were for (4,3,3) and (4,4,3) vicinal surfaces. The barriers for jumps rather than atom exchange proved much higher and are therefore not given. At straight steps of a four row stripeshaped island the barriers were also higher. From an fcc site at an A-type edge the activation energy EB proved to be 0.36 eV, and from an hcp site 0.34 eV. For a B-type edge from an fcc site a barrier of 0.19 eV had to be overcome, and 0.22 eV from an hcp site. It is interesting that the barrier for exchange was higher for stripe islands but the barrier for hopping was exactly the same. Leonardelli et al. [127] also did estimates for the behavior of nickel atoms on Pt(111). For vicinal surfaces, for an atom coming from an fcc site and encountering an A-type edge, the barrier EB was 0.37 eV, versus 0.39 eV when originally at an hcp site. For B-type edges the barriers EB were comparable, 0.32 eV and 0.30 eV, and the step-edge barrier Ea proved to be 0.09 eV for B-type steps and 0.17 for A-type steps. At straight edges of four row-striped islands the activation energy was somewhat higher, EB=0.41 eV and 0.45 eV at A-type steps and 0.35 eV versus 0.33 eV at B-type steps. Again exchange was more favorable than hopping. Comparisons with experiments are not available here. Descent at a kink site was less demanding than descent at straight steps. For both atoms, Pt and Ni, the descent at a kink in step B had the lowest barrier, lower than for movement of the adatom on the flat surface. However the strange observations of Pt atoms in four fold sites suggest that the data may not be reliable. Information about descending platinum steps is quite wide, but disagreement between direct measurements and theoretical calculations should be clarified. Also quite surprising is that theory predicted the exchange mechanism on both types of steps.

478

Diffusion in special environments

6.2.20

Gold There are only a few investigations about descending motion over gold steps, started by Stoltze [41] with effective medium theory. For jumps over steps at the gold (100) plane, an adatom needed EB = 0.546 eV, while the value for exchange was comparable at 0.528 eV. Interlayer motion was slightly more demanding than movement on the terrace, which required ED = 0.490 eV. For Au(110), Stoltze looked at atom interlayer jumping in the [001] direction and found a value EB of 0.675 eV; for exchange at this step the barrier is slightly higher, 0.691 eV. Interlayer exchange in the ½1
10 direction required EB = 0.582 eV. Jumping on the Au(110) terrace in the ½1
10 direction was much less demanding at ED = 0.265 eV, but jumps along [100] required 0.67 eV. Exchange on the terrace in the [100] direction is more probable, occurring over 0.554 eV. On the Au(111) only one type of step was investigated and it turned out that jumping was more favorable than exchange, with a barrier EB of 0.260 eV compared to 0.326 eV. Movement on the flat (111) surface also proceeded by hopping with an energy ED of 0.102 eV. Li and DePristo [42,43] calculated barriers to surface diffusion with the corrected effective medium theory, CEM, in order to explain homoepitaxial growth. The barriers to surface diffusion over B-type step edges are listed in Table 6.2. Movement on the flat surface required ED = 0.029 eV, overcoming a step cost more, EB = 0.090 eV, but stepping down at the kink involved a barrier EK of 0.047 eV, while descending from a 19 atom island took 0.034 eV. These findings have to be treated with caution. The interlayer movement of adatoms over gold steps needs extensive investigation – it is impossible to draw any conclusions based on the data available.

6.2.21

Lead Zhu et al. [33] carried out density-functional estimates of lead atoms moving over lattice steps on Pb(110). The adatom undergoes exchange with the lattice, but this involves an activation energy of 0.46 eV for an in-channel ascent, and 0.37 eV for the reverse process. For a cross-channel ascent, already shown in Fig. 6.21, the barrier was 0.43 eV, with a barrier of the same height for descent. An investigation of lead steps was also done by Li et al. [128] in 2006. They compared the growth of a cluster with the decay of a vacancy island and found a new method to determine the step-edge barrier. From STM observations of Pb on Pb(111) they deduced an additional step-edge barrier Ea of 0.083 ± 0.060 eV. They named this an effective barrier, since it was an average for descending A-type steps, B-type steps, as well as at kinks. Based on EAM calculation they derived the barrier Ea for hopping over an A-type step as 0.1 eV via a rolling-over motion (jump), and the barrier over B steps as 0.03 eV via concerted motion, probably exchange. The effective barrier derived from experiments lies between the EAM values. The mechanism of descent is also in agreement with what was found for Ir(111) – A-type hopping and B-type exchange.

6.3 Atom lifetime versus the step-edge barrier

6.3

479

Atom lifetime versus the step-edge barrier Having examined the energetics of atom incorporation at steps it is now worthwhile to briefly look at the detailed kinetics of adatom incorporation and its dependence on the potential landscape near the descending step. The effects of step-edge barriers on diffusion were discussed in 1997 by Kyuno and Ehrlich [129], who envisioned the edge potentials shown in Fig. 6.56. The lifetime τi of an atom at site i before escaping is described by τ i ¼ 1=ðλi þ μi Þ þ λi =ðλi þ μi Þτ iþ1 þ μi =ðλi þ μi Þτ i
1 ; l5u:

(6:5)

Here λi represents the rate of jumping to the right and μi the rate to the left for an atom at position i. The rate of escape, either from site l or u, will depend upon the rate ν0B expð
EB =kTÞ at which the final barrier EB is surmounted. However, atoms can also move back into the plane interior, giving for the rate of escape from an interior site 1=τ x  ν0B expð
EB =kTÞ=f1 þ b
1=2Þ  exp½
ðER
ED Þ=kTg;

(6:6)

where b indicates the width of the plane and the barrier ER preventing an adatom from returning to the center is indicated in Fig. 6.56. When next to the edge there is a trap which keeps atoms in a deep asymmetric well, then the rate of return jR will be negligible and the rate of escape can be described by 1=τ x  jB ¼ ν0B expð
EB =kTÞ: μi

λi

ED

i

jD

EB

ER

(b)

Site

(6:7)

Barrier

i+1 jD jR

ED

jB

ER

(a)

EB Trap

x=

Fig. 6.56

b -2

b-1

b

Energy barrier E as well as jump rates j in diffusion over a descending step [129]. (a) Potential for trap at end. (b) For repulsive barrier with adjacent deep well.

480

Diffusion in special environments

Another extreme case occurs when there is a repulsive barrier at the edge but the rate of return jumps exceeds the jump rate in diffusion, jR jD . Atoms at the edge equilibrate with the interior and the rate of escape can be written as 1=τ x  ½ν0B =ðb
1=2Þ exp½
ðEB
ER þ ED Þ=kT:

(6:8)

It is clear that the lifetime of atoms before incorporation depends upon the detailed energetics close to the edge. In Fig. 6.57 we present a comparison of the effect of a conventional reflective barrier, a trap as in Fig. 6.56b and a barrier with a trap shown in Fig. 6.56a on the rate of adatom escape from a plane with a radius of seven spacings. The plots are from Monte Carlo simulations with an input energy for terrace diffusion of ED = 0.3 eV, a step-edge barrier of EB = 0.52 eV, and a standard frequency factor 1012 sec−1. From the plot it is clear that the presence of a trap next to the step influences the rate of adatom escape, as well as the energy and prefactor derived from an Arrhenius plot. Up till now the barrier to diffusion over the step edge has been emphasized. However, barriers are also possible away from the edge, for example on Pt(111) and Ir(111), and such barriers influence the rate of incorporation on those planes. The influence of such interior barriers on the lifetime of an atom was investigated by Oh et al. [104]. On a onedimensional plane the lifetime of an atom in position i has already been given by Eq. (6.5). For an atom deposited at the center of a 1D cluster without additional traps and barriers it was found that the lifetime τc was τ c ¼ ðb2 þ 1Þ2 =ð2jD Þ;

(6:9)

where b2 gives the size of the cluster. For a traditional reflective barrier at the edge, shown in Fig. 6.58a, the lifetime was given by two terms,   1 2 (6:10) τ c ¼ b2 =ð2jD Þ þ b2 þ =jB2 ; 2

Rate of Escape (sec–1)

101

Trap ν0B = 1.0 × 1012sec–1 EB = 0.52 eV ER = 0.61 eV

10–1 Barrier

ν0B = 3.0 × 1011sec–1 10–3

EB = 0.51 eV ER = 0.35 eV Conventional

ν0B = 1.3 × 1011sec–1

10–5

EB = 0.52 eV ER = 0.30 eV 10–7 3.5

4

4.5

5

5.5

6

6.5

7

1000/T (K–1) Fig. 6.57

Dependence of atom incorporation rate at descending step upon the reciprocal temperature [129]. Trap corresponds to potential curve in Fig. 6.56a, Barrier to Fig. 6.56b, Conventional to Fig. 6.58a.

481

6.3 Atom lifetime versus the step-edge barrier

jB2 jD (a) Step-Edge Barrier

jD EB2

ED

x=0

jB1

b2

jD

jD (b) Interior Barrier

EB1

ED

x=0

b1

ED EB1

ED

x=0 Fig. 6.58

b2 jB1

jD (c) Interior Barrier & Trap

ED

b1

EB2

b2

Possible potentials at edge of 1D cluster [104]. Only right half of cluster is shown.

where the first term described the time to reach the edge site, the second the lifetime for incorporation starting at that point. The influence of the size of the cluster on the lifetime is shown in Fig. 6.59, where the lifetime to incorporation increases by a factor of ~ 30 due to the presence of the step-edge barrier. Barriers not at the step edge but in the interior have a more dramatic effect. What has to be taken into account is that an atom, after overcoming the interior barrier, still has a chance to return and continue movement in the interior. The lifetime is now described by   1 τ c ¼ b21 =ð2jD Þ þ b1 þ =jB1 þ τ b1 þ1 ; (6:11) 2 where the additional term τ b1 þ1 gives the lifetime starting just beyond the interior barrier, which was given by τ b1 þ1 ¼ ðb2
b1 Þ½ðb2
b1 þ 1Þ=ð2jD Þ þ ðb1 þ 1=2Þ=jB1 :

(6:12)

Investigated was the influence of an interior barrier at different positions on a cluster of 20 atomic spaces. For starts at the center, the lifetime increased more than 5 times depending on the location of the interior barrier, as shown in Fig. 6.60. The presence of an outermost trap in the cluster (Fig 6.58c) in addition to the interior barrier is not felt strongly by an atom and the lifetime overlaps with the lifetime for a single interior barrier, shown in Fig 6.60 by solid circles. However, the additional step-edge barrier has

482

Diffusion in special environments

4

0.2 1D

τc (sec)

3

0.15 Step-Edge Barrier

2

0.1

1

0.05 No Barriers

0 0

Fig. 6.59

5

10 15 Cluster Radius b2

0 20

Lifetime τc for atom starting at center to incorporate into step of 1D cluster of atoms [104]. Diffusion barrier 0.26 eV, step-edge barrier 0.33 eV, T = 140 K.

20

Pt Model

1D

τ (sec)

15

τb1 + 1 10

τc Interior Barrier

5

Step Edge 0

Fig. 6.60

0

5 10 15 Location of Interior Barrier b1

20

Effect of interior barrier at b1 on lifetime for incorporation on a 1D cluster with radius of 20 nearest-neighbor spacings. [104]. Solid points give lifetime starting at center. Dashed horizontal indicates lifetime just in presence of step-edge barriers.

significant influence on the lifetime of the atom, indicated by solid diamonds (or by Pt model) in Fig 6.60; the lifetime is around 20% higher. For atoms deposited at random positions, which is a more realistic picture for vapor deposition, the lifetime is given in Fig. 6.61, with the position of the interior barrier shifting in the direction of the step edges, since atoms deposited close to the edge have a shorter lifetime than when deposited at the center. For interior barrier positions between 12 and 2 the random start can lower the lifetime by one fifth or more. The effect is the strongest for interior barrier positions midway between the center and edges of the cluster. The lifetime was also investigated for a 2D cluster. On such a cluster with a reflective step-edge barrier the lifetime, averaged over all positions, is described by 5τ 4 ¼ ðb2 þ jD =jB2 Þb2 =ð2jD Þ:

(6:13)

483

6.3 Atom lifetime versus the step-edge barrier

20

1D Pt Model

(sec)

15

τc 10 Interior Barrier 5 Step Edge 0

Fig. 6.61

0

5 10 15 Location of Interior Barrier b1

20

Lifetimes for atoms starting at random on 1D cluster with radius of 20 nearest-neighbor spacings compared with lifetime starting in center (in grey) [104].

2D

τc (sec)

1.6

0.08 Step-Edge Barrier

0.8

0.04 No Barriers 0

0 0

5

10

15

20

Cluster Radius b2 Fig. 6.62

Comparison of lifetime of atom starting at center on a 2D circular cluster with and without a step-edge barrier [104]. EB2 = 0.33 eV, ED = 0.26 eV, T = 140 K.

The dependence of the lifetime, with and without a step-edge barrier, on the cluster size is given in Fig. 6.62. On the circular cluster the lifetime is roughly half that on the 1D cluster and follows the same trend. It is clear the lifetime is influenced by both the step-edge barrier and the cluster size. For a cluster of 20 atomic spacings an interior barrier significantly influences the lifetime of atoms deposited at the center, as in Fig. 6.63 An interior barrier between 4 and 18 increases the adatom lifetime at least three fold compared to the lifetime with only a step-edge barrier. For random starts of the atom the lifetime is cut by more than 25% when the interior barrier is located at < 14, compared with central starting, as is clear in Fig. 6.64. However, the interior barrier still increases the overall lifetime of atoms. On 2D clusters lifetimes are smaller than on 1D clusters, but the qualitative trends are the same. The presence of interior barriers has an effect not just on atom lifetime for incorporation. Oh et al. [130] also showed such barriers increase the rate of dimer nucleation.

484

Diffusion in special environments

12 Pt Model

τc (sec)

10 8

2D

6 4

Interior Barrier Step Edge

2 0

Fig. 6.63

0

5 10 15 Location of Interior Barrier b1

20

Effect of interior barrier on lifetime to incorporation starting from the center of a 2D cluster with a radius of 20 nearest-neighbor spacings [104].

12 Pt Model

(sec)

10 8

2D

6

τc

4 Interior Barrier Step Edge

2 0 0

Fig. 6.64

5 10 15 Location of Interior Barrier b1

20

Effect of internal barrier on lifetime of atoms starting at random positions on a 2D cluster [104].

Fig. 6.65 gives the fraction nucleated on a square of half-width 20, with an internal barrier of 0.33 eV and a diffusion barrier of 0.26 eV. As the internal barrier is moved toward the plane edges, there is a very significant increase in nucleation, as collisions between adatoms increase. The interior barrier together with the step-edge barrier cause a faster decrease in the rate of nucleation at increasing temperature. What emerges from these studies is that the detailed energetics over a cluster have a really noteworthy effect on diffusion and also on surface reactions.

485

6.4 Comparisons

1 0

Fraction Nucleated

0.8

2

4 Interior Barrier

0.6 6 0.4

8 10

0.2

0 50

δD

No Additional Barrier

100

150

200

250

300

350

400

T (K) Fig. 6.65

Effect of interior barrier on nucleation of dimers on square (100) plane with half-width 20 [130]. Interior barrier height 0.33 eV, diffusion barrier 0.26 eV, no additional barrier at the step edge; δD gives distance between interior barrier and step edge.

6.4

Comparisons The amount of experimental data available for step-edge barriers is not small, but not all of it is consistent. It is therefore useful to summarize some of the results in Tables 6.5 through 6.7. What is clear from the interpretation of growth experiments for Ag(111) in Table 6.5 is that the results for the additional step-edge barrier Ea are all reasonably close together, regardless of the particular analysis, except for the work of Roos et al. [79], which is drastically lower. Unfortunately, there are no direct studies of atom loss over step edges available for comparison. The situation is different for studies of the step-edge barrier on Pt(111), given in Table 6.6, where deductions are listed from experiments. These are interesting for their considerable spread. Markov’s [108] early findings are unusual as they are so large compared to other results. It is not clear that such a spread in the additional step-edge barrier is connected with the different experiments analyzed or just due to a different method of analysis. Noteworthy is the small barrier found in direct observations of atom loss from a Pt(111) plane [111]. It is worth noting that the direct observations by Kyuno and Ehrlich took into account the existence of an interior barrier on this plane. The other findings considered only a traditional step-edge barrier. As shown by Oh et al. [104], an interior barrier can have significant influence on the lifetime of an atom on the cluster and will influence the growth characteristics. However, it is still not clear if interior barriers will be present for straight steps. Interior barriers might be responsible for obtaining different values of the additional step-edge barrier for the

486

Diffusion in special environments

Table 6.5 Summary of step-edge barriers Ea for self-diffusion on Ag(111) derived from experiments. Step-edge barrier Reference

Ea(eV)

Vrijmoeth et al. [73] Smilauer and Harris [35] Meyer et al. [74] Bromann et al. [75] Markov [45] Roos et al. [79] Morgenstern et al. [85]

0.150 ± 0.030 0.13 − 0.23 0:150þ0:040
0:020 0.120 ± 0.015 0.15 0.040 0.13 ± 0.04

Table 6.6 Step-edge barriers derived in experiments for self-diffusion on Pt(111). Step-edge barrier Reference

Ea (eV)

Smilauer and Harris [35] Meyer et al. [74] Markov [108] Markov [45] Kyuno and Ehrlich [111] Krug et al. [112]

0.15–0.27 0:165þ0:050
0:020 0.31, 0.44 ≈ 0.21 0.06 ± 0.01 0.12

Table 6.7 Total calculated barrier for escape of Pt from Pt(111). Total escape barrier EB (eV) Reference

Step A

Step B

Kink A

Kink B

Villarba and Jónsson [116] Jacobsen et al. [120] Li and DePristo [42] Feibelman [121] Ma´ca et al. [126] Leonardelli et al. [127]

0.30 (Ex) 0.41(Jump)

0.18 (Ex) 0.37 (Ex) 0.044 (Ex) 0.64(Ex) 0.40(Ex) 0.17(Ex)

0.08 (Ex)

0.06 (Ex) 0.26 (Ex) 0.012(Ex)

0.42(Ex)

0.26(Ex)

0.31 (Ex) 0.50 (Ex) 0.28(Ex)

same material on analyzing different experiments. That is why the detailed analysis of experiments is so important. There is as large a difference in the various calculated barriers shown in Table 6.7 for crossing a step edge on Pt(111), but some of the simpler efforts are in excellent agreement with experiments. Theory clearly distinguishes

6.5 Atom descent over many-layered steps and between facets

487

between types of steps and suggests an acceleration of interlayer movement at kinks, which has not been observed experimentally so far. The mechanism of descent is still not uncovered; however, exchange seems to be the most likely. Theoretical results for the barrier to atom incorporation on Ir(111) are in reasonable agreement, and the experimental work is in good agreement with the results for Pt(111). Unfortunately, there is a factor of two difference between the barriers from theory and experiment. Surprisingly, theory is quite good at predicting the value for the diffusion barrier on a flat surface for this system and clearly predicts easier descent at kinks, a fact not yet confirmed by experiment. There still stays open a question about the mechanism of descent on Ir(111). Exchange was shown experimentally to be involved in descent of a hetero-atom from steps of type B. Unfortunately, experiments cannot distinguish between a jump and exchange for self-diffusion, while theory offers a confusing image for B-type steps, since one study suggests exchange as the leading mechanism [106] while another favors a simple jump [107]. There is a wealth of theoretical investigations for Cu(111) but no reliable experimental values have been derived so far for additional step-edge barriers, they are scattered from 0.113 to 0.224 eV. Scatter in theoretical values is in the same range from 0.04 to 0.14 eV for steps of type B and from 0.15 to 0.4 for A-type steps. For B steps, theory seems to agree that descent will proceed by exchange, while for steps A the situation is not resolved. Steps for fcc(100) surfaces have received less attention from both experiment and theory. There is no clear understanding if descent would proceed by hopping or exchange. Half of the investigations prefer the latter, the other half the former. Most investigated is the densely packed step. Descent over the step is usually associated with a negative barrier; in practice this means that if the temperature is high enough to promote movement, such steps will not exist at the surface. On fcc(110) surfaces the easier descent is usually in the direction along the channels of this plane, in most cases by atom jumps. The calculational efforts have been useful for revealing interesting features, but it is clear that detailed experimental information is really needed to put diffusion over step edges on a sound footing. The mechanism of descent has not always been clear. Quite interesting is the prediction of a negative step-edge barrier; this finding will definitely have to be verified by experiment. A survey of the available data on step-edge barriers is provided in Table 6.8.

6.5

Atom descent over many-layered steps and between facets Up to this point, atoms have been assumed to descend over monoatomic steps. However, this is not always the case. It is possible that the descending movement over multilayer steps or over whole facets has to be taken into account as well in the detailed description of interlayer movement. Here we will consider a few investigations tackling this problem. Unfortunately study in this area is limited to

488

Diffusion in special environments

Table 6.8 Additional step-edge barriers. Experiment

Theory

System

Type of step

Mechanism Ea (eV)

Ea (eV)

Method

Ref.

Al/Al(111)

A§

Al/Al(111)

B§

Ex/Jump Ex Ex/Jump Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex

0.04/0.41 0.26& 0.02/0.29 0.02& 0.02& ~0.10& ~0.09& ~0.20& ~0.13& ~0.33& ~0.41& ~0.4& ~0.03&

DFT LDA MS DFT LDA MS MS MS MS MS DFT MS DFT MS MS Nucl MC RHEED Sim STM MC STM MC STM MC EMT EAM EAM EAM CEM CEM EAM EMT EMT EAM EAM EMT EAM EMT EAM EMT EAM Nucl Nucl STM STM Rean STM EMT 2nd Moment DFT F-S EAM

[27–29] [32] [27–29] [32] [32] [32] [32] [32] [33] [32] [33] [32] [32] [35] [36] [37] [38] [38] [38] [41] [40] [12] [40] [42,43] [42,43] [40] [41] [44] [12] [12] [44] [12] [41] [40] [41] [12] [45] [45] [48] [49,50] [48] [41] [51] [62] [55] [57]

Al/Al(100)

Al/Al(110)

/{110} /{100} /{110} /{111} /{100} /{111} /{110}

0–0.1 0.03–0.06 0.03–0.04 0.41 0.34 0.36

Fe/Fe(100)

Fe/Fe(110)



Ni/Ni(111) A B

Ni/Ni(100)

BK

K Ni/Ni(110)





Jump/Ex Ex/Jump Ex Ex/Jump Ex Ex Ex/Jump Jump/Ex Ex/Jump Jump Jump Ex/Jump Ex/Jump Jump/Ex Ex/Jump Ex/Jump Jump

0.461/0.601 0.494/0.529 0.18 0.113/0.449 0.058 0.031 0.22/0.28 0.229/0.332 0.113/0.289 0.34 0.09 − 0.109/0.064 0.3/0.36 0.273/0.403 0.87/0.93 0.51/0.771 0.94 0.113 0.080–0.055 0.116 ± 0.002 0.224 ± 0.009 0.19 ± 0.002

Cu/Cu(111)

K A

Jump/Ex Ex Ex Jump/Ex Ex/Jump

0.312/0.397 0.057 ~0.62–0.81& 0.4/0.43 0.251/0.481

489

6.5 Atom descent over many-layered steps and between facets

Table 6.8 (cont.) Experiment System

Type of step

K



A AK B BK

Pd/Pd(111)

Pd/Pd(100)

Ref.

Jump Ex/Jump

0.15 0.24/0.46 0.3 0.171/0.371 0.137 0.057/0.462 0.14/0.4 0.056/0.471 0.38 0.04/0.46 0.06 0.281/0.361 0.083 0.148 0.045 0.147/0.201 0.20/0.36 ~0.11& 0.179/0.232 0.13/0.32 0.185 0.05/0.28 0.03/0.29 0.09/0.33 0.03 − 0.15/0.08 ~0&/~0.12& 0.023/0.043 − 0.16/0.08 − 0.16/0.07 0.48/0.93 0.414 0.398/0.548 ~0.57& ~0.29& 0.188/0.282 0.34 0.28 0.32/0.58 0.32 0.24/0.59 0.09 0.202/0.374 − 0.004 − 0.029 0.070/0.195

MC EAM MD EAM EAM CEM EAM F-S EAM MC EAM MD EAM EAM CEM DFT DFT EMT EMT F-S EMT RGL EAM EAM EAM EAM 2ndMoment EAM F-S EMT EAM EAM EAM EAM EMT DFT DFT EMT MS MS RGL RGL RGL RGL EMT CEM CEM EMT

[61] [64] [63] [57] [42,43] [54] [55] [57] [61] [64] [63] [57] [42,43] [67] [66] [41] [68] [55] [44] [69] [70] [57] [54] [64] [51] [57] [55] [44] [57] [64] [54] [70] [41] [33] [33] [41] [131] [131] [71,72] [71,72] [71,72] [71,72] [41] [42,43] [42,43] [41]

Ex/Jump Ex Jump Ex Jump/Ex Ex/Jump Jump Jump/Ex Ex/Jump Jump Ex/Jump Ex/Jump Ex/Jump Ex Ex/Jump Jump/Ex Ex/Jump Ex/Jump Ex/Jump Ex/Jump Jump Ex/Jump Ex Ex Jump/Ex Ex



Rh/Rh(111)

Method

BK

Cu/Cu(100)

Co/Cu(110)

Ea (eV)

Ex/Jump Ex Ex/Jump Ex/Jump Ex/Jump Jump Ex/Jump

B

Cu/Cu(110)

Mechanism Ea (eV)

AK B

Fe/Cu(111)

B BK

Theory

Ex/Jump Ex Ex/Jump Ex Jump/Ex Ex Ex Jump/Ex

490

Diffusion in special environments

Table 6.8 (cont.) Experiment

Theory

System

Type of step

Mechanism Ea (eV)

Ea (eV)

Method

Ref.

Pd/Pd(110)



Jump/Ex Ex/Jump

0.054/0.292 0.411/0.413

EMT EMT STM Nucl STM STM Nucl Nucl Nucl STM EMT CEM STM CEM CEM STM CEM STM

[41] [41] [73] [35] [74] [75] [45] [45] [79] [85] [41] [42] [86] [42] [42,43] [86] [42] [75]

MC MC MD EAM MD EAM Sim, STM Sim, STM EMT EMT DFT GGA DFT LDA RGL EAM FDB EAM VC EAM FDB EAM VC EMT EMT DFT RGL DFT RGL EMT RGL FIM FIM FIM FIM

[88] [88] [92] [92] [95] [96] [41] [44] [93] [93] [69] [94] [94] [94] [94] [44] [41] [33] [97] [33] [97] [41] [97] [98] [21,22] [21,22] [16]

0.150 ± 0.030 0.13–0.23 0:15þ0:040
0:020 0.120 ± 0.015 0.15 0.12 0.04 0.13 ± 0.04

Ag/Ag(111)

A

Jump/Ex Ex/Jump

AK B

Ex Ex/Jump

BK

Ex

0.295/0.391 0.29/0.42 0.16 0.17 0.044/0.41 0.08

Au/Ag(111)

A B

No Inter Inter Ex Ex/Jump



K



K Pd/W(110) Ta/W(110) W/W(110)

0.068 ± 0.001 0.074 ± 0.002 − 0.051 to − 0.041 − 0.055/0.614 0.03 0.07

Ag/Ag(100)

Ag/Ag(110)

0.03 0.030 ± 0.005

Ag/Ag island / Pt(111)

Jump/Ex Jump/Ex Ex/Jump Ex/Jump Ex/Jump Jump/Ex Ex/Jump Ex/Jump Ex/Jump Jump/Ex Jump/Ex Ex Jump Ex Ex/Jump Ex/Jump Ex

0.085/0.194 0.114/0.149 ~0/0.10 ~0/0.18 0.12/0.16 0.11/0.16 0.03/0.22 − 0.1/0.03 − 0.17/0.07 0.034/0.037 0.069/0.246 ~0.24& 0.07 ~032& 0.28/0.52 0.314/0.355 0.10 0.27 ± 0.02 − 0.05 0.36Æ ~0.065

491

6.5 Atom descent over many-layered steps and between facets

Table 6.8 (cont.) Experiment System

Type of step

Mechanism Ea (eV)

Re/W(110)

Ir/W(110)

Pt/W(110) Pd/W(111) Re/W(321) Pd/W(211) Re/Re(0001) W/Re(0001) Ir/Ir(111)

Jump Jump A AK B BK AV BV

Ir/Ir(100) Rh/Ir(100) Ni/Pt(111)

A B

Pt/Pt(111)

Jump/Ex Jump/Ex Jump/Ex Jump/Ex Ex/Jump Jump/Ex Jump/Ex Jump/Ex Jump/Ex Jump/Ex Ex Jump Ex/Jump Ex/Jump Jump/Ex

Theory Ea (eV)

0.03 0.51Æ − 0.05 0.228 ± 0.009œ 0.199 ± 0.009Æ − 0.10 0.27Æ 0.087 0.145 ± 0.006Æ 0 ~0.54Æ 0.297 0.195 ± 0.012Æ − 0.02 0.52Æ 0.61 ± 0.06 0.05 0.29 ± 0.08 0.20 ± 0.028 0.35 ± 0.021 0.20 ± 0.03 0.66/1.34 0.77/1.55 0.63/1.30 0.53/1.21 0.54/0.78 0.67/0.79 0.63/0.70 0.53/0.61 0.51/0.58 0.51/1.13 − 0.02 ± 0.08 0.04 ± 0.08 0.17/0.44 0.09/0.44 0.204/0.311 0.15–0.27 0:165þ0:05 0:020 ~0.21 0.06 ± 0.01 0.10

A

0.12 0.15 0.25 0.17hcp0.22fcc/ 0.62–0.72

Jump Ex/Jump 0.31

Method

Ref.

FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM RGL RGL RGL RGL RGL RGL RGL RGL RGL RGL FIM FIM EAM EAM EMT Nucl Nucl Nucl FIM FIM LAM TDT EMT EAM

[21,22] [21,22] [23] [23] [23] [21,22] [21,22] [23] [23] [21,22] [21,22] [23] [23] [21,22] [21,22] [98] [19] [98] [100] [100] [101] [107] [106] [106] [107] [106] [107] [106] [107] [107] [107] [101] [101] [127] [127] [41] [35] [74] [108] [111] [111] [112] [112,114] [120] [116]

Nucl

[108]

492

Diffusion in special environments

Table 6.8 (cont.) Experiment System

Type of step

AK B

Theory

Mechanism Ea (eV)

Ea (eV)

Method

Ref.

Ex/Jump Ex/Jump Ex/Jump Ex Ex Ex

0.02/0.24 0.33/0.35 0.20/0.74 0.00 0.25 0.10hcp

VASP RGL EAM EAM RGL EAM Nucl EMT CEM VASP RGL EAM EAM EMT CEM RGL EMT EMT EMT EMT CEM CEM EMT EMT EMT STM EAM EAM DFT DFT

[121] [71,72,126] [127] [116] [71,72,126] [116] [108] [120] [42,43] [121] [71,72,126] [127] [116] [120] [42,43] [71,72,126] [41] [41] [41] [41] [42,43] [42,43] [41] [41] [41] [128] [128] [128] [33] [33]

0.44

B BK

Ex Ex Ex/Jump Ex/Jump Ex/Jump Ex Ex Ex Ex Jump/Ex Ex Jump/Ex Jump/Ex Ex Ex Ex/Jump Ex/Jump Ex

A B

Jump Ex Ex Ex

BK

Pt/Pt(100) Pt/Pt(110)



Au/Au(111)

Au/Au(100) Au/Au(110)

0.083 ± 0.060

Pb/Pb(111)

Pb/Pb(110)

0.21 − 0.004 0.35/0.51 0.23/0.36 0.08/0.73 − 0.02 0.10 − 0.036 0.09 0.066/0.108 0.362 0.524/0.874 0.158/0.224 0.061 0.018 0.038/0.056 0.41/0.426 0.317 0.1 0.03 ~0.17& ~0.23&

&

The activation energy for terrace diffusion ED used was based on values in Chapters 4 and 5; These are Al(100) 0.25 eV, Al(110) 0.3 eV, Al(111) 0.04 eV, Cu(100) 0.4 eV, Cu(111) 0.04 eV, Cu(110) 0.25 eV, Ag(110) 0.3 eV, Pb(110) 0.2 eV; Æ assumed kT/h; § A-type steps on the fcc(111) plane are sometimes designated by /{100}, being the orientation of the step and {100} its faceting; similarly B-type steps are sometimes shown as /{111}; œ assumed D = 3.6 × 10–3cm2/sec

theoretical effort. As first shown by Valkealahti and Manninen [132] using effective medium theory, diffusion between facets of aluminum polyhedra occurred from (111) and (100) planes. For jumping between (100) and (111) facets, the static value of the energy was 0.47 eV while the dynamic one amounted to 0.48 eV. The opposite jumps, from (111) to (100), cost a bit less, only 0.28 eV for the static value and 0.30 eV for the dynamic estimate. Jumps between (111) planes had a comparable value, 0.28 eV for the static and 0.29 eV for the dynamic barrier. Exchange between two (111) facets

6.5 Atom descent over many-layered steps and between facets

493

turned out to be less demanding, requiring only 0.17 eV. They also looked at jumps with “pull,” which means hopping across the edge with the help of a nearby facet. Such jumps with “pull” turn out to have quite a low energy. For moving between (111) facets only 0.11 eV, and between (111) and (100) facets with “pull” cost 0.31 eV. Chain exchange, between (111) facets, involving a row of atoms from the (100) plane, also had a very low energy, which increased with the number of atoms involved in a process. For three atoms, the energy required was only 0.08 eV, with four atoms 0.17 eV, and with a chain of five atoms 0.28 eV. The diffusion mechanism and energetics between (111) and (100) facets of gold and silver were examined with RGL potentials using the nudged elastic band method by Baletto et al. [133]. They worked with two polyhedras with 1289 atoms (25 atoms on the (100) plane) and 201 atoms (9 atoms on the (100) plane). Adatom jumps from the (111) plane to the (100) cost 0.36 eV on Ag clusters and 0.34 eV on Au clusters. Exchange between the same planes required slightly less energy, 0.32 eV for silver and 0.24 for gold. Jumping between two (111) planes was comparable and cost 0.34 eV for silver and 0.29 eV for gold; exchange was more favorable, with 0.23 eV for silver and 0.15 eV for gold. They also observed a chain process which covered movement to (111) planes through the (100) plane, a process illustrated in Fig. 6.66a. This cost only 0.19 eV for silver and 0.21 eV for gold. Multiple exchange, named a “half-chain” process, shown in Fig. 6.66b where the adatom pushes some atoms in a row and an atom is emitted on the (100) plane a few atomic spaces apart from the original entrance of the adatom. Such a process cost 0.38 eV for silver and less, only 0.29 eV, for gold. Based on their finding they concluded that diffusion on and from (100) facets took place only at high temperatures due to the presence of low-energy long-range mechanisms which bypass this plane.

Fig. 6.66

Mechanisms for atom motion over Ag and Au polyhedra. (a) Chain process between opposite (111) facets. Atom pushes a row until end atom falls onto (111). (b) Half-chain for transfer from (111) to (100) facet (after Baletto et al. [133]).

494

Diffusion in special environments

Table 6.9 Step-edge barriers by exchange (in eV) for aluminum [32]. Number of Layers Initial Facet

Step Orientation/Final Facet

1

Multi

{111}

/{100} /{110} /{111} /{100} /{110} /{111} /{100} /{110} /{111}

0.30 0.06 0.06 0.35 0.34 0.45 0.63 0.33 0.70

0.30 0.04 0.21 0.25 0.35 0.68 0.72 0.47 0.83

{100}

{110}

Fig. 6.67

Multilayer step, a {100} surface, between two {111} planes of aluminum (after Liu et al. [32]).

In 2001 Liu et al. [134] examined the movement of adatoms between two facets of aluminum. For jumps between two {111} planes an energy of 0.45 eV was required, while exchange between these two faces needed only 0.22 eV. For jumps from the {111} to the {100} plane, an atom had to overcome a barrier of 0.48 eV, while exchange again was less demanding, requiring only 0.30 eV. Jumps between two {100} planes cost 0.60 eV, exchange only 0.25 eV. Jumps from a {100} to a {111} plane needed 0.78 eV, exchange 0.68 eV. What is interesting in this investigation is that exchange seemed to be generally less demanding than hopping. Liu et al. [32] in 2002 used molecular statics to discuss a new topic – atom descent occurring over a step made up of many layers, as shown in Fig. 6.67 for diffusion from the (111) plane of an fcc crystal. For aluminum they made estimates, in Table 6.9, giving the diffusion barriers over monoatomic steps compared with multilayers for diffusion by atom exchange.3 From this study it can be concluded that diffusion across a 4-layer step can be treated as the movement from one type of facet to another, since the energy of crossing saturates at this thickness. It appears that multilevel transitions are sometimes more favorable than descent over a single layer step. The same year, Huang [135] introduced the step-facet barrier for the intersection of two /{111}

3

The orientation of the steps appears first, followed by a description of the steps faceting.

6.5 Atom descent over many-layered steps and between facets

495

Table 6.10 Step-edge diffusion barriers (in eV) on Cu(111) [63]. Multilayer

Monolayer

Fig. 6.68

{111} → {111} {111} → {100} {100} → {111} {100} → {100} /{111} /{100}

0.33 0.40 0.67 0.20 0.10 0.34

Two Cu(111) surfaces separated by multilayer (111) step. Dark spheres (indicated by arrow) represent different stages of adatom migration: initial, transition, and final state (after Huang et al. [63]).

steps on a Cu(100) surface. The energy for such crossing, however, was quite high, 0.98 eV, making such a process insignificant. This work was continued by Huang et al. [63] with embedded atom calculations [136] for multilayer steps from copper (111), as in Fig. 6.68. Their results, in Table 6.10, reveal that monolayer steps allow easier passage of copper atoms than do multilayer structures. However, multilayer descents are associated with energetics which should be achievable at higher temperatures. Diffusion on Cu(111) in their estimates required overcoming a barrier of 0.04 eV, in good agreement with experiments. They also introduced indium as a surfactant and, based on the difference in sublimation energies, estimated that the bond for indium was about 71% of the copper-copper bond. Huang et al. postulated an 84% reduction of the step-edge barrier due to the presence of indium on the surface, that is a lowering of the stepedge barrier from 0.33 eV to 0.28 eV. Carrying out growth experiments of copper as well as of copper with indium, they showed that the copper layer grew much smoother when indium was present, supporting their claim that indium lowered the step-edge barrier.

496

Diffusion in special environments

Fig. 6.69

Self-diffusion events over multilayer steps between Al(100) planes. Energy barriers for diffusion calculated with VASP program are indicated in the figure (after Buatier de Mongeot et al. [137]).

In an attempt to explain their atomic force microscope observations of huts, Buatier de Mongeot et al. [137] in 2003 did density-functional calculations in the generalized gradient approximation of VASP for aluminum atoms diffusing from Al(110) over multistep facets, as shown in Fig. 6.69. The energetics of diffusion over steps were comparable to what had been found for cross-channel transitions on flat Al(110), namely 0.49 eV. What was surprising was that the outer-corner crossing required more energy than inner-corner crossing, and movement from (111) to (100) facets was preferred over the opposite, making the multilayer barrier asymmetric. In 2004, calculations with the VASP package were done by Zhu et al. [33] to examine atom diffusion at edges of fcc(110) planes. They also considered transitions from fcc(110) over facets or ridges, displayed in Fig. 6.70. The energetics for these steps are listed in Table 6.11, and it is clear that these processes do not involve overcoming very high barriers and will influence interlayer motion. What is interesting is that an in-channel climb onto a (100) facet was energeticaly favorable by exchange, but for a cross-channel climb onto a (111) facet atomic hopping was preferred. Calculations with EAM potentials were carried out by Wang et al. [64] to characterize transitions down steps from (100) and (111) copper surfaces. Their findings for monoatomic steps are given in Table 6.12, and just as seen previously for other materials, some of the barriers are not too high. The descent from the (111) plane via a /{111} step cost much less than descending over a /{100} step, 0.08eV compared to 0.28 eV. On a flat (111) surface a moving atom overcame a barrier of 0.04 eV. Both steps were descended by exchange rather than hopping. However, on a Cu(100) surface, an atom moved by hopping according to the authors, with an energy of 0.48 eV. Descent proceeds by exchange, overcoming a barrier of 0.57 eV on the /{111} step and

497

6.5 Atom descent over many-layered steps and between facets

Table 6.11 Self-diffusion barriers (in eV) calculated with VASP program for ridge crossing on (110) surface in Fig. 6.70 [33]. Al

A → B Ex B → A Ex B → E Jump E → B Jump C → D Ex D → C Ex

Cu

In-channel

Cross-channel

In-channel

Cross-channel

0.18 0.69 0.65 0.40 0.33 0.13

0.03 0.70 0.36 0.38 0.69 0.04

0.19 0.47 0.33 0.24 0.34 0.11

0.03 0.55 0.42 0.37 0.43 0.03

Table 6.12 Diffusion barriers calculated with EAM potentials for Cu adatoms on copper [64].

Fig. 6.70

On (111) surface

Barrier

On (100) surface

Barrier

Flat surface Down step /{111} Down step /{100} Step /{100}→/{111} /{111} → /{100}

0.04 Jump 0.08 Ex 0.28 Ex

Flat surface Down step /{111} Down step /{100}

0.48 Jump 0.57 Ex 0.32 Ex

0.40 Ex 0.42 Ex

/{111}→/{100} /{100} → /{111} /{111} → /{111} /{100} → /{111}

0.55 Jump 0.85 Jump 0.57 Jump 0.56 Ex

Facets and ridges between fcc(110) planes for (a) in-channel and (b) cross-channel directions (after Zhu et al. [33]).

498

Diffusion in special environments

Table 6.13 Facet–facet diffusion barriers (in ev) calculated for Cu adatoms with EAM potentials for different number of layers N [64]. N≥4

N= 1 Process

Ex

Jump

Ex

Jump

{111} → /{111} {111} → /{100} {100} → /{111} {100} → /{100}

0.08 0.28 0.57 0.32

0.50 0.50 0.81 0.55

0.30 0.40 0.75 0.17

0.57 0.57 0.82 0.55

0.32 eV on the / step. The barrier on the latter step was lower than for movement on the terrace. Barriers for transitions from one facet to an adjacent one are listed in Table 6.13; some of these processes again involve crossing reasonably small barriers. The authors also considered multilayer transitions, but these are generally of a much higher energy. Transitions over many-layered steps have so far been explored in calculational studies, and it should be interesting to compare these with experimental results once these become available.

6.6

To ascending step edges In the past, the standard view was that atoms diffuse randomly over a surface until they strike an ascending lattice step and incorporate there. A direct look at this process was taken by Wang [138] in 1993. He created a small iridium cluster of 12 atoms on top of the Ir(111) plane, and deposited an iridium adatom on the terrace nearby the cluster, then observed the movement of the adatom, which is shown in Fig. 6.71. The atom diffused on the plane for a while and eventually incorporated into the central Ir12 cluster, but without appearing at sites in the immediate vicinity of the cluster. The combination of several deposition experiments is given in Fig. 6.72, which clearly reveals an empty zone around the Ir12 in the center. This effect was also noted for larger clusters, containing up to twenty-one atoms. One possible way of accounting for this behavior is to invoke a small potential barrier around the cluster which keeps atoms from coming closer. This idea has been tested by depositing an atom on a terrace with a cluster on it, and then comparing the probability of incorporating into the cluster with the probability of being trapped close to the edge of the (111) plane. Results at different temperatures are given in Fig. 6.73, and show no temperature dependence for the ratio of incorporated atoms to atoms trapped at the edge. This suggests the absence of an additional barrier around the cluster. Instead, the activation energy for atom motion in the vicinity of the island has to diminish, so that atoms are sucked into the cluster. From these experiments it was estimated that a diminution of the normal activation energy for diffusion by ~10% could account for these results.

6.6 To ascending step edges

499

Fig. 6.71

Ir atom incorporation into ascending step of Ir cluster on Ir(111) [138]. (a) Cluster formed by field evaporation. (b) Ir atom has been deposited on (111) plane. (c)–(k) Atom moves over (111) surface after heating. (l) Atom has incorporated into cluster after 118 diffusion intervals.

Fig. 6.72

Location of Ir atom after diffusion on Ir(111) surface with Ir12 cluster at the center, showing empty region around cluster [138].

Additional insight into this phenomenon was obtained in further experiments, in which Wang [139] deposited iridium atoms onto an Ir(111) plane kept at ≈ 20 K with a cluster in the center. At this temperature deposited adatoms were immobile on a flat cluster free surface; however, as shown in Fig. 6.74, for a surface with a cluster, atoms are found all over the surface, except in a ring around the cluster, with the ring of roughly the same size as in the diffusion experiments. In this example, the experiment was done with a central cluster of 12 iridium atoms, but the same sort of empty zone was found when the cluster size was increased to 59 atoms, as in Fig. 3.136c. The one way of accounting for this behavior at low temperatures, where thermal diffusion does not occur, is to assume the potentials at sites around the cluster are seriously perturbed, as indicated in Fig. 3.137,

500

Diffusion in special environments

= 0.91 ± 0.05

Ni /Nt

1

0.1

7.5

8.0

8.5

9.0

9.5

10.0

1000/T (K–1) Fig. 6.73

Ratio of Ir atoms incorporated into central Ir12 cluster, Ni, compared to Nt, the number trapped at the edge of the plane [138]. Lower right: sketch of potential in edge vicinity consistent with experiments.

24 16

y( 0/2)

8 0 –8 –16 –24 –48

2

–36

–24

–12

0

12

24

10

36

48

x(√3/6 0) Fig. 6.74

Distribution of Ir atoms deposited on Ir(111) plane at ≈ 20 K, with Ir12 cluster at the center [139]. Empty zone surrounds cluster.

much more than was necessary to understand the experiments exploring diffusion to the cluster. Theoretical estimates have now been made showing such deformations in the immediate vicinity of a step. For example, Villarba and Jónsson [116], working with EAM interactions, plotted the energy of a platinum atom approaching an ascending A- as well as a B-type step on Pt(111). As is clear in Fig. 6.75, the well adjacent to the step disappears due to interactions. Villarba and Jónsson in Fig. 6.75 also found decreased barriers for an atom going from site 5 to 4 at the type A step and between site 4 and 3 for a B-type step. Their results, like the results of Wang [139], show that once an atom reaches site 3 it can easily hop towards the island and incorporate there. This view of the

501

6.6 To ascending step edges

(a) 8–7 4–3 9 6–5 2–1

type A step

2–3 6–7 1 4–5 8–9

type B step

(b) 0.5 type A step

Energy (eV)

0.0

3

0.5

4

5

2

8

9

type B step

0.0 1

7

6

2

3

4

5

6

7

8

9

–0.5 Diffusion to step edge Diffusion atop flat terrace

–1.0 1 0

2

4

6

8

10

12

14

Displacement (Å) Fig. 6.75

Deformation of normal surface potentials at ascending A- and B-type steps on Pt(111), estimated using EAM interactions. (a) Hard-sphere model of steps on fcc(111). (b) Potential energy plots at two types of steps (after Villarba and Jónsson [116]).

incorporation of atoms into ascending steps has been criticized by Kellogg [140], who labeled the proposed energetics around an island as unphysical. However, at the moment all the evidence points to this view as being correct. Atom approach toward ascending Ni steps was investigated by Liu and Adams [40] using EAM potentials and molecular statics. They found forbidden regions next to both descending and ascending steps on the Ni(111) plane. On approaching an ascending step of type B, the energy barrier rose to 0.06 eV three spacings away from the step. Close to the step, the diffusion barrier dropped suddenly to 0.04 eV. When one space apart from the ascending step the adatom is very unstable and usually reaches the step by a double jump and incorporates. A movement of clusters with up to five atoms on the Ni(111) plane showed similar effects. Approaching an A-type step, adatoms experienced the same situation, only the energy barriers were different. Jumps to the forbidden region cost the same energy 0.06 eV, while near the step, the decrease of energy was around 0.022 eV. The forbidden region was probably caused by stress around the step. Hontinfinde et al. [97], who did molecular dynamics using RGL potentials on a Ag(110) surface, noticed slightly lower barriers approaching a step from the lower terrace than for movement on the terrace. Recently Smirnov et al. [65] using ab initio and MC simulations examined the interactions of Cu adatoms with Cu(111) islands. They found concentric repulsive

502

Diffusion in special environments

Fig. 6.76

Incorporation of Ir atom into Ir chain on W(110) plane [141]. (a) FIM image of chain with one Ir atom nearby. (b) Trajectory of Ir atom after diffusion at 360 K for 5 sec intervals. Atom never approaches side of chain.

rings of 25 meV around islands at 4–7 Å, an attractive ring of −5 meV at 8–14 Å and repulsive ring of 2 meV at 15–20 Å. Based on their MC simulation they claim that at low temperatures adatoms prefers to stay around 9 Å away from the island. They associate this effect with quantum confinement of electrons at the surface close to the step. However overcoming the repulsive barrier should be temperature dependent and that is not what was observed by Wang for Ir(111) indicating a different origin of the empty zone for this system. Somewhat related to the approach of diffusing atoms toward ascending steps is their behavior in the vicinity of one-dimensional iridium and palladium chains on W(110), which has been examined by Koh and Ehrlich [141]. When an iridium atom is deposited on a surface on which there is such a chain, and the surface is heated, the atom does not incorporate into the long side of the chain. Instead, as shown in Fig. 6.76, the atom diffuses parallel to the chain but at some distance from it until it finally attaches itself to the chain end. When there is a second chain present on the plane, an iridium adatom again avoids the chain sides, as in Fig. 6.77, and on incorporation goes to the end of the chain. Enough is known about interactions between iridium adatoms to make an estimate of how the potential energy looks around the chains. That is shown in Fig. 6.78, and it is clear that going to the chain end is the lowest energy path. Palladium adatoms deposited between palladium chains behave quite similarly despite a large difference in binding energy of Pd and Ir. However, due to differences in second and third nearest-neighbor interactions, as well as trio interactions, palladium is expected to attach to the side of a chain when it reaches it. Once a Pd adatom reaches the side of the chain it than moves very rapidly along this chain to attach at the end (something never observed for Ir adatoms). This is an example of how long-range interactions alter the normal diffusion behavior; in the absence of chains, atom movement on bcc(110) occurs equally along all directions. The problem of long-range interactions will be discussed in detail in Chapter 10. Similar effects have more recently also been discovered by Stepanyuk et al. [142], who looked at the diffusion of Co atoms on Cu(111) by scanning tunneling microscopy. What they found is shown in Fig. 6.79 – the formation of metastable one-dimensional chains of Co atoms. Interactions between Co atoms had been examined in both theory and

503

6.6 To ascending step edges

Fig. 6.77

Trajectories for single Ir atom movement between two Ir chains on W(110) after 10 sec heating interval at 340 K [141].

150 100 Free Energy (meV) 50 0 –50 5

–100

4 3

–150 1 2 3 4 5 6 η[111] Fig. 6.78

7 6

2 7 8

-

χ[111]

1 9 10 11

Interaction free energy for Ir adatom and two Ir chains on W(110) at η = 0 and η = 12 [141]. Estimates based on previously measured pair- and trio-interactions.

experiments, as shown in Fig. 6.80. This allowed Stepanyuk et al. to calculate interactions between a Co atom and a Co chain, as illustrated in Fig. 6.80. Approaching the chain from the side requires overcoming a higher diffusion barrier than does moving parallel to the chain, so that atoms incorporate into the chain ends. These profound morphological effects are brought about by interactions of only a couple of meV, illustrating the importance of long-range effects in growth phenomena.

504

Diffusion in special environments

Fig. 6.79

STM image of Co atoms deposited on Cu(111) surface at 11 K at a coverage of 0.006 ML. Long one-dimensional chains are formed, due to long-ranged interactions (after Stepanyuk et al.[142]).

Most recently, Negulyaev et al. [143] have carried out STM observations of the diffusion of copper adatoms near copper chains on the Cu(111) plane. This work is analogous to the earlier FIM observations by Koh and Ehrlich [141] on iridium adatom motion near Ir chains and Pd atoms migrating near Pd chains on W(110). In the latter study the interactions were obtained from experimental determinations of the interaction between adatoms, whereas Negulyaev et al. resorted to DFT to evaluate them. Next to a single chain of Cu atoms, the copper adatom diffused parallel to the chain at a separation of ~25 Å. For a copper adatom between two copper chains separated by 55 Å diffusion again occurred parallel to the chains. Estimates of the linear density of states from calculations and from experiment are given in Fig. 6.81, and reveal small oscillations. Interactions between chain and adatom are strongly attractive at a nearest-neighbor distance from the chain followed by repulsion at a larger separation preventing incorporation. Negulyaev et al. operated at 12 K, at which the copper jump rate was ~1/200 that of the iridium on W(110) [141], which could eliminate the random excursions of the Ir adatom. In any event, the effect of longrange interactions near steps and chains is now quite well established. In 2005, Mo et al. [66] undertook density-functional calculations of iron diffusion on stepped copper(111) surfaces. At a descending step, the barrier to diffusion and incorporation of the iron adatom by exchange with the substrate amounted to 0.070 eV. For incorporation at an ascending step the barrier to exchange was much higher, and amounted to 0.66 eV. The displaced Cu atom in an ascending exchange process jumps down over a barrier of 0.34 eV. The second Fe atom approaching the ascending step joins the first Fe atom embedded in the step. The authors predict that a similar mechanism will also be observed for W atoms at a Cu(111) surface. Ding et al. [67] also made DFT estimates for the same system, and for iron incorporating at a descending step the barrier amounted to ≈ 0.173 eV, as is indicated in Fig. 6.82. At

6.6 To ascending step edges

Fig. 6.80

505

Effects of long-ranged interactions on formation of one-dimensional Co chains, calculated for Co adatom and chain of Co atoms. Shown is the energy barrier, and separately the path of an incorporating atom. (a) Co atoms in the chain positioned at first nearest-neighbor distance. (b) Co atoms positioned at first minimum in long-range potential. In both, incorporation only occurs at end of chain (after Stepanyuk et al.[142]).

an ascending step, the barrier to incorporation was much smaller, and was only 0.020 eV. They observed an oscillatory dependence of adatom–step interactions for both descending and ascending steps with a period of about 1.5 nm. They associated this effect with the electron-charge distribution at the step. It should be noted here, however, that the possibility of atom exchange occurring was not mentioned, and this could account for these high values compared to what was found by Mo et al. Only little has so far been done to characterize the approach of atoms to ascending step edges, and it is clear that more work is needed to arrive at a clear view of these phenomena. Right now it appears, however, that the standard picture of randomly

506

Diffusion in special environments

(b) dl/dV (arb. units)

(a)

0

50

60

24 4 Distance

0 –4 × 0.005 1

(a) 2

3

Distance(nm)

Interaction energy (meV)

174

28

0

Fig. 6.82

20 30 40 Distance (Å)

Diffusion of single Cu adatom between two chains of Cu atoms on Cu(111). (a) STM image of adatom and chains, separated by 55Å. Temperature 12 K. (b) Calculated and measured LDOS at right angle to chains (after Negulyaev et al. [143]).

Interaction energy (meV)

Fig. 6.81

10

170 4 Distance

0 –4 –8

(b) 0

1

2 3 Distance(nm)

Energy of interaction between Fe atom on copper steps on Cu(111) plane, derived from DFT calculations. (a) At ascending step. (b) At descending step. Energy change for incorporation is much higher for ascent (after Ding et al. [67]).

diffusing atoms striking islands will have to be modified to allow more generally for the interactions exerted on the adatoms.

6.7

Diffusion near dislocations In the theory of crystal growth a breakthrough occurred in 1951 when Burton, Cabrera, and Frank [144] showed that a screw dislocation emanating from a surface, as depicted in Fig. 6.83, could rapidly bring about incorporation of atoms deposited on the surface, forming a growth spiral. Much work at atomic resolution was carried out in the next decades [145], but nothing at all was done to characterize the diffusion of atoms over the surface as influenced by such dislocations. From Fig. 6.83 it seems clear that a screw dislocation might have effects similar to those of a lattice step, and in the literature, growth has been represented as occurring via atom diffusion just as on a normal surface, with atoms eventually incorporating at the ascending lattice step [146].

6.7 Diffusion near dislocations

507

Fig. 6.83

Schematic showing formation of growth spiral by addition of atoms to a screw dislocation emerging in (a) from the surface.

Fig. 6.84

End of tip intercepted by screw dislocation [147]. (a) Schematic of surface, with numbers indicating the height. (b) He field ionization image of screw dislocation, with numbers corresponding to schematic. (c) Image after field evaporation of four layers. Edge atoms in (b) and (c) can be superposed, indicating core is perpendicular to the surface.

Recently, however, Antczak and Józwik [147] carried out the first atomic scale observations of atom behavior on a dislocated surface, using the field ion microscope to examine a tungsten atom on a W(110) plane intersected by a screw dislocation. An image of the well-annealed surface is shown in Fig. 6.84 before and after field evaporation to remove four atomic layers. The image obtained after evaporation was directly superposed on the original image, proving that the defect was indeed primarily a screw dislocation. When a tungsten atom was evaporated onto this surface and its locations after two hundred ninety 10 sec diffusion intervals at 340 K were recorded, the results, in Fig. 6.85a were surprising. The ½1
11 grid through the atom positions near the dislocation core appeared tilted to the right at the lower ramp of the dislocation with respect to similar lines at the left on the upper ramp of the dislocation, suggesting a possible change in interatomic spacings arising from stress. However, the ½
111 grids from the lower and upper ramps match each other quite nicely in Fig. 6.85b.
Furthermore,pffiffithe ffi 2mean-square displacement in the ½111 direction amounted to
4.00 ± 0.49 (ap l ffiffiffi3/2) , while in the ½111 direction it was considerably smaller, only 0.93 ± 0.12 (al 3/2)2. Distortion in the ½1
11 direction created an easy path for the adatom to move over the surface, while on the undefected surface the mean-square displacement

508

Diffusion in special environments

End 300

300

End

250

y

y

250

200

200

Start

Start 150 250

Fig. 6.85

(b)

150

(a) 300

350 x

400

450

[110] [111] [111]

250

300

[001]

350 x

400

450

Location of surface sites on W(110) at which tungsten adatom was observed after diffusion at 340 K [147]. (a) Grid, associated with the ½1
11 direction is tilted at the right with respect to the left. (b) Grid associated with ½
111 direction does not reveal any tilt. edge atoms

outer region (O)

upper ramp (U)

core

core

p ndin g ste

asce

asce

ndin

g ste

p

inner region (I) lower ramp (L)

(a) Fig. 6.86

(b)

Schematic of different regions of the dislocation loop [147]. (a) Upper and lower ramps. (b) Inner and outer regions, the former out to ~ six atom spacings from dislocation core.

for both the directions were equal. Most interesting, however, are the details of the atomic displacements in diffusion in different locations. On an ordinary flat W(110), diffusivities were measured for three separate sets of 100 observations of a tungsten atom, and were found to be close to each other: 0.206 ± 0.030, 0.175 ± 0.030, and 0.168 ± 0.021, all in units of (al/2)2 per second. On the dislocated surface, atomic behavior varied depending upon the location on the surface, and it proved useful to subdivide the atomic transitions, as indicated in Fig. 6.86, between lower (L) and upper ramps (U), as well as between regions closer (I) and farther from the core (O). As is clear from the diffusivities in Table 6.14, diffusion on the outer region on the lower ramp (LO) of the loop is roughly five times that on the upper loop (U), where it amounts to only 64% the normal value. In the inner region (I) , close to the core of the dislocation, the adatom moved approximately 2.5 times slower than in the outer region. Finally, the adatom does

509

References

Table 6.14 Diffusivity of adatom at 340 K on W(110) intersected by screw dislocation and on undefected W(110) plane [147]. t Type of surface Defect-free Screw Dislocation



Region

N

(sec) ðal =2Þ2

All data (O) (I) (U) (L) (OU) (OL) (IU) (IL)

1200 290 140 135 123 167 68 72 42 91

4 10 10 10 10 10 10 10 10 10

pffiffiffi ðal 2=2Þ2

1.353 ± 0.065 1.503 ± 0.079 4.583 ± 0.531 4.459 ± 0.483 6.500 ± 1.003 6.071 ± 0.864 2.556 ± 0.357 2.763 ± 0.476 2.081 ± 0.378 1.659 ± 0.236 6.425 ± 0.852 6.521 ± 0.785 2.162 ± 0.531 1.397 ± 0.235 10.597 ± 1.758 10.486 ± 1.492 1.048 ± 0.202 0.952 ± 0.190 3.220 ± 0.507 3.440 ± 0.670

Dx

Dy

ðal =2Þ2 sec
1

pffiffiffi ðal 2=2Þ2 sec
1

0.169 ± 0.008 0.229 ± 0.027 0.325 ± 0.051 0.128 ± 0.018 0.104 ± 0.019 0.321 ± 0.043 0.108 ± 0.027 0.530 ± 0.088 0.052 ± 0.010 0.161 ± 0.026

0.188 ± 0.010 0.223 ± 0.024 0.304 ± 0.044 0.138 ± 0.024 0.083 ± 0.012 0.326 ± 0.039 0.070 ± 0.012 0.524 ± 0.075 0.048 ± 0.010 0.172 ± 0.034

not appear to find incorporation into the outer descending lattice step advantageous – it stays two spacings away. A full explanation for the various observations is not yet available. What appears to be happening, however, is that the dislocation introduces strain into the surface, above and beyond that predicted by elastic theory, and this affected the diffusion in entirely unexpected ways. From this study strain parallel to the surface can be expected as indicated by the lateral changes in atomic distances visible in the ½1
11 grids. There are also differences in movement between upper and lower ramps probably associated with strain perpendicular to the surface. The strain also appears to change from tensile to compressive with distance from the core. Clearly much more will have to be done to establish possible contributions from edge components of the dislocation and to determine if the distribution observed here is general for other materials as well. What is clear already, however, is that adatom diffusion is very strongly affected by the presence of the dislocation and cannot be approximated as being similar to atom behavior on an ordinary surface. It is also clear that novel, unexpected phenomena are still being discovered in the field of surface diffusion.

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516

Diffusion in special environments

[136] Y. Mishin, M. J. Mehl, D. A. Papaconstantopoulos, A. F. Voter, J. D. Kress, Structural stability and lattice defects in copper: Ab initio, tight binding, and embedded-atom calculations, Phys. Rev. B 63 (2001) 224106 1–16. [137] F. Buatier de Mongeot, W. Zhu, A. Molle, R. Buzio, C. Boragno, U. Valbusa, E. G. Wang, Z. Zhuang, Nanocrystal formation and faceting instability in Al(110) homoepitaxy: True upward adatom diffusion at step edges and island corners, Phys. Rev. Lett. 91 (2003) 016102 1–4. [138] S. C. Wang, G. Ehrlich, Adatom motion to lattice steps: A direct view, Phys. Rev. Lett. 70 (1993) 41–44. [139] S. C. Wang, G. Ehrlich, Atom condensation at lattice steps and clusters, Phys. Rev. Lett. 71 (1993) 4174–4177. [140] G. L. Kellogg, Experimental observation of ballistic atom exchange on metal surfaces, Phys. Rev. Lett. 76 (1996) 98–101. [141] S. J. Koh, G. Ehrlich, Self-assembly of one-dimensional surface structures: Long-range interactions in the growth of Ir and Pd on W(110), Phys. Rev. Lett. 87 (2001) 106103 1–4. [142] V. S. Stepanyuk, A. N. Baranov, D. V. Tsivlin, W. Hergert, P. Bruno, N. Knorr, M. A. Schneider, K. Kern, Quantum interference and long-range adsorbate-adsorbate interactions, Phys. Rev. B 68 (2003) 205410 1–5. [143] N. N. Negulyaev, V. S. Stepanyuk, L. Niebergall, P. Bruno, W. Hergert, J. Repp, K. H. Rieder, G. Meyer, Direct evidence for the effect of quantum confinement of surfacestate electrons on atomic diffusion, Phys. Rev. Lett. 101 (2008) 226601 1–4. [144] W. K. Burton, N. Cabrera, F. C. Frank, The growth of crystals and the equilibrium structure of their surfaces, Phil. Trans. Roy. Soc. A 243 (1951) 299–358. [145] K. M. Bowkett, D. A. Smith, Field Ion Microscopy (North-Holland, Amsterdam, 1970), Chapter 5. [146] B. Mutaftschief, The Atomistic Nature of Crystal Growth (Springer-Verlag, Berlin, 2001), Section 17.3. [147] G. Antczak, P. Jóźwik, Atom movement on a dislocated surface, Langmuir 24 (2008) 9970–9973.

7

Mechanism of cluster diffusion

So far we have concentrated on the behavior of single atoms. However, when several atoms are present on a plane and they diffuse, atoms may collide with each other and form a cluster. Such events are illustrated in Figs. 7.1, 7.2, and 7.3 [1–3], where coalescence of two as well as three atoms is observed directly using the field ion microscope. These clusters are of considerable interest for the roles they play in the growth and dissolution of a crystal, as well as their effects on surface chemical reactions. Of primary concern here is the ability of atom clusters to diffuse over a crystal surface, and it is this aspect of cluster properties that we will emphasize. We will look at the conditions under which a cluster moves as a whole and also when parts of a cluster start moving independently. Different mechanisms of diffusion will be discussed in some detail. In probing the diffusion of clusters, it is worthwhile to distinguish two different types of mechanisms – movement by single atom jumps, and by concerted atom displacements. In the first category, five types of movement have so far been identified, which are: 1. diffusion by sequential atom jumps (Fig. 7.4a), 2. peripheral displacements (Fig. 7.4b), 3. by the leapfrog mechanism (Fig. 7.4c), 4. by the correlated evaporation– condensation mechanism also known as detachment–attachment (Fig. 7.4d) or terrace limited diffusion, and 5. by the evaporation–condensation mechanism (Fig. 7.4e), in which one atom leaves a cluster and then a different atom from the terrace attaches to the cluster. Concerted displacements recognized so far are illustrated in Fig. 7.5. They may occur by sliding of the cluster as a whole (Fig 7.5a), by dimer or trimer shearing shown in Fig 7.5b or by reptation in Fig 7.5c, and finally by motion of a dislocation (Fig. 7.5d). Recently a new concerted motion was discovered. Trushin et al. [4], during investigations of Cu clusters on a Cu(100) surface using EAM, noticed an interesting mechanism contributing to movements of large clusters – internal dimer rotation, illustrated in Fig. 7.6. Internal dimer rotation differs from isolated dimer rotation since it involves concerted motion of all cluster atoms at the moment of transition. We will start with a rather simple example, the diffusion of a dimer, made up of two atoms, moving in one dimension. This will provide a formal but simple introduction to brief presentations of the motion of larger clusters. Only after this outline of the formalism will we discuss the investigations, both experimental as well as theoretical, carried out to examine processes involved in cluster diffusion.

518

Mechanism of cluster diffusion

Fig. 7.1

Formation of Re2 on W(211) plane [1]. In (a), three rhenium atoms have been deposited on the plane, with two in one channel. Warming for 30 sec to 286 K induces motion in (b)–(e), and ends in creating an in-channel dimer.

Fig. 7.2

Creation of Pt2 cluster on Pt(111) plane [2]. In (a) a single Pt atom has been deposited on the surface maintained at ~ 30 K. Another atom has been deposited in the center of the plane in (b). Five second intervals of heating to ~ 100 K bring about atom motion in (c)–(e). Dimer, which starts to diffuse at  150 K, has been formed in (f).

Fig. 7.3

Formation of rhenium trimer on W(110) plane [3]. In (a) three Re atoms have been deposited on a cold (~ 20 K) surface. (b) After 10 sec at 420 K, atoms have moved, and after another such heating interval have combined into a triangular trimer in (c).

Mechanism of cluster diffusion

519

Fig. 7.4

Schematics of single atom events in cluster diffusion. (a) Sequential displacements, top view. (b) Peripheral diffusion, top view. (c) Leapfrog movement, side view. (d) Detachment–attachment mechanism, top view. (e) evaporation–condensation.

Fig. 7.5

Schematics showing concerted mechanisms of cluster motion. (a) Gliding. (b) Shearing. (c) Reptation. (d) Via Dislocation.

520

Mechanism of cluster diffusion

Energy E(eV)

0.6 0.5 0.4 0.3 0.2 0.1 0.0

0

10 20 Configuration

30

Fig. 7.6

Potential energy calculated for internal Cu dimer rotation in hexamer on Cu(100) using EAM interactions. Schematics at the border show intermediate states of dimer rotating in hexamer (after Trushin et al. [4]).

7.1

Via single atom jumps

7.1.1

One-dimensional movement of dimers At the start we confine our attention to one-dimensional motion of a dimer with one atom in each of two adjacent diffusion channels, as on the (211) plane of a bcc crystal, and will focus on the displacement of the center of mass (COM). What we present is based largely on work by Reed [5], but an alternative presentation, based on analogy with electrical networks, has been given by Titulaer and Deutch [6]. Dimer movement is sketched schematically in Fig. 7.7. At the top of the diagram are shown the two configurations of the dimer, designated by 0 when the two atoms are in line, so that the dimer is straight, and by 1 when the dimer is slanted. The spacing of sites in the channel is taken as ‘0 , the nearest-neighbor spacing, so that when one atom in the dimer jumps, the center of mass moves a distance ‘0 =2, which will be the unit of length for now. In state 0, the straight state, an atom can jump to the right at the rate a, and to the left at the rate d. In state 1, an atom jump to the right occurs at the rate b, and to the left at the rate c. Rates of jumps to the right at position x can more generally be written as lx , and the rate to the left, again from position x, as μx . Just as we did for the motion of single atoms, we can write out the Kolmogorov relation for dimers as dpx =dt ¼ lx
1 px
1
ðlx þ μx Þpx þ μxþ1 pxþ1

x ¼ 0; 1; 2; . . .

(7:1)

where px is the probability of having the center of mass at position x. On multiplying this equation by x and summing over all positions, we obtain the differential equation governing the average of x. The average is given by X xpx ; (7:2) hxi ¼ x

521

7.1 Via single atom jumps

0

1

0

1

0

1

–2

–1

0

1

2

3

λx

a

b

a

b

μx

d

c

d

c

Type

0

x

Fig. 7.7

–4

–3

b

a d

c

4

5

Diffusion of cross-channel dimer represented as a random walk [5]. Dimer configurations are indicated for center of mass position x. Heavy lines show unit cell. lx gives rate of center of mass motion to the right, μx to the left.

so that dhxi=dt ¼

X

lx
1 px
1 x


x

X

ðlx þ μx Þpx x þ

x

X

μxþ1 pxþ1 x

(7:3)

x

¼ hðx þ 1Þlx i
hxlx i þ hðx
1Þμx i
hxμx i

(7:4)

dhxi=dt ¼ hlx i
hμx i:

(7:5)

and finally

Integrating over time we find that hxi ¼ ðhlx i
hμx iÞt;

(7:6)

as at time t = 0, the mean displacement is equal to zero. Henceforth it will be useful to indicate the state of the dimer at the starting site by the superscript z, where z can be either 0 or 1. The probability of the center of mass being at a site of type 0 is given by X ðzÞ P0 ¼ pk k ¼ 0; 2; 4; . . .: (7:7) k

Similarly, for being at a site of type 1, the probability is X ðzÞ P1 ¼ pk k ¼ 0; 1; 3; 5; . . .

(7:8)

k

furthermore ðzÞ

ðzÞ

P0 þ P1 ¼ 1; as there are only two types of sites present, type 0 and 1. The rate constants averaged over all positions may be written as X ðzÞ ðzÞ px lx ¼ aP0 þ bP1 hlx i ¼ x

(7:9)

(7:10)

522

Mechanism of cluster diffusion

hμx i ¼

X

ðzÞ

ðzÞ

px μx ¼ dP0 þ cP1 :

(7:11)

x

The mean displacement for our dimer from Eq. (7.6) therefore becomes h i ðzÞ ðzÞ hxi ¼ ða
dÞP0 þ ðb
cÞP1 t;

(7:12)

and the mean displacement will in general have a finite value as time increases. It is now a straightforward matter  to work  out the variance of the displacement, that is the displacement fluctuation Δx2 ¼ x2
hxi2 , since we already know the value of the average of x, 〈x〉. Following the procedures used to obtain 〈x〉, we multiply the probability px by x2 and sum over all values of x to get  2 X 2 x px ; (7:13) x ¼ x

so that X X X  d x2 =dt ¼ lx
1 px
1 x2
ðlx þ μx Þpx x2 þ μxþ1 pxþ1 x2 x

x

(7:14)

x

D E  E  D ¼ ðx þ 1Þ2 lx
x2 lx þ ðx
1Þ2 μx
x2 μx

(7:15)

¼ hlx ð2x þ 1Þi
hμx ð2x
1Þi:

(7:16)

For the time derivative of the displacement fluctuation we have   d Δx2 =dt ¼ d x2 =dt
2hxidhxi=dt and therefore  d Δx2 =dt ¼ hð2x þ 1Þlx i
hð2x
1Þμx i
2hxihlx i þ 2hxihμx i ¼ 2ðhxlx i
hxμx iÞ þ hlx ið1
2hxiÞ þ hμx ið1 þ 2hxiÞ:

(7:17)

(7:18)

In evaluating the displacement fluctuation, we will limit ourselves here to symmetrical dimer diffusion, in which the jump rate d to the left is equal to the rate a to the right, and the rate c to the left equals the rate b, so that in accord with Eq. (7.12) the mean displacement vanishes. Under these conditions, X hxlx i ¼ xlx px ¼ ahxi0 þbhxi1 (7:19) x

hxμx i ¼

X

xμx px ¼ ahxi0 þbhxi1

(7:20)

x

Here hxii denotes the value of the position x averaged over all sites of type i. The differential equation for the displacement fluctuation now appears as

523

7.1 Via single atom jumps

 d Δx2 =dt ¼ 2½ahx0 i þ bhx1 i
ðahx0 i þ bhx1 iÞ þ ðaP0 þ bP1 Þð1
2hxiÞ þ ðaP0 þ bP1 Þð1 þ 2hxiÞ ¼ 2ðaP0 þ bP1 Þ:

(7:21)

ðzÞ

The probability P0 is readily accessible by summing the probability pk over all sites of type 0, so it follows from Eq. (7.1) that X X X lx
1 px
1
ðlx þ μx Þpx þ μxþ1 pxþ1 x ¼ 0; 2; 4;    dP0 =dt ¼ x

x

x

(7:22) For the general case of unsymmetric motion we find dP0 =dt ¼ bP1
ða þ dÞP0 þ cP1 ¼ ðb þ cÞP1
ða þ dÞP0 :

(7:23)

Since P1 = 1 − P0 dP0 =dt þ ða þ b þ c þ dÞP0 ¼ b þ c:

(7:24)

Solving this differential equation we obtain P00 ¼

1 ½b þ c þ ða þ dÞ exp
ða þ b þ c þ dÞt; aþbþcþd

(7:25)

inasmuch as at t = 0, P00 = 1. If now we again restrict ourselves to symmetrical walks with a = d and b = c, then P00 ¼

1 ½b þ a exp
2ða þ bÞt aþb

(7:26)

P01 ¼

a ½1
exp
2ða þ bÞt: aþb

(7:27)

In a similar series of steps we find b ½1
exp
2ða þ bÞt aþb

(7:28)

1 ½a þ b exp
2ða þ bÞt: aþb

(7:29)

P10 ¼ P11 ¼

Inserting these two results for starting at position 0 in the differential equation for the displacement fluctuation, Eq. (7.21), we see that  0 2a d Δx2 =dt ¼ ½2b þ ða
bÞ exp
2ða þ bÞt: aþb After integration, this transforms to    2 0 2a ða
bÞ 2bt þ ð1
exp
2ða þ bÞtÞ : Δx ¼ aþb 2ða þ bÞ

(7:30)

(7:31)

524

Mechanism of cluster diffusion

If instead we insert the expressions appropriate for starting at position 1 we obtain  1 2b ½2a
ða
bÞ exp
2ða þ bÞt; d Δx2 =dt ¼ aþb

(7:32)

and on integrating get 

  1 2b ða
bÞ 2at
ð1
exp
2ða þ bÞtÞ ; Δx2 ¼ aþb 2ða þ bÞ

(7:33)

all of which is in agreement with what was found by Reed [5]. Our concern is primarily with longer distance diffusion, that is with long diffusion times. Under these conditions, the second term in Eqs. (7.31) and (7.33) becomes vanishingly small, and the displacement fluctuation simplifies to  2 4abt Δx ¼ ; aþb

(7:34)

and is independent of the starting point. For long diffusion times the probability of being at a particular type of site also simplifies, and we have P0 ¼

b aþb

P1 ¼

a : aþb

(7:35)

Under these circumstances, local equilibrium prevails for the dimer configurations, so that P0 a ¼ P1 b;

(7:36)

and from the ratio of the occupation probabilities we can obtain the ratio of jump rates P1 a ¼ : P0 b

(7:37)

So far we have used the displacement ‘0 =2 as the unit of length. If, as is more usual, we resort to standard units then 

ab a Δx2 ¼ ‘0 2 t ¼ ‘0 2 t: aþb 1 þ a=b

(7:38)

It is obvious that from diffusion measurements of the center of mass we do not find out the jump rates  a and b. However, by combining determinations of the displacement fluctuation Δx2 with measurements of the ratio P1 =P0 it is simple to ascertain the individual jump rates. Here we note that in position 1 two configurations are possible, whereas at 0 positions only one can exist. For the rate a we therefore have a ¼ 2a expð
Ea =kTÞ;

(7:39)

b ¼ b expð
Eb =kTÞ:

(7:40)

and for b

7.1 Via single atom jumps

Fig. 7.8

525

Tungsten dimer diffusion on W(211) [8]. (a) Hard-sphere model of W(211) plane with crosschannel dimer in straight and staggered configuration. (b)–(e) FIM images of tungsten dimer diffusing on W(211) plane at 255 K.

It is worth noting that expressions for the displacement fluctuation could alternatively have been obtained from the moment generating function, as was done for the diffusion of single atoms. Derivations of this type have been given by Wrigley et al. [7], but do not yield much more than has been found here by direct calculations. We also want to emphasize that these derivations have been important for giving us Eqs. (7.37) and (7.38), which have been instrumental in analyzing data on dimer diffusion. Movement of cross-channel dimers can be easily observed in the FIM, where both dimer atoms are recognizable and the configuration is readily observed. Diffusion of a tungsten cross-channel dimer on the W(211) plane was reported in 1974 by Graham and Ehrlich [8]. Their data, illustrated in Fig. 7.8, clearly showed that diffusion occurred by single atom transitions. Let us now look at the diffusion of an in-channel dimer, as was first done by Reed [1]. Such a dimer may be able to move as a whole, in a concerted jump, or else by two jumps of single atoms. The latter mechanism is illustrated in Fig. 7.9, and the displacement fluctuation is given by Eq. (7.38), derived for the diffusion of cross-channel dimers. There is, however, a difference in the rate constant a of the in-channel dimer. This lacks the factor of two present for cross-channel dimers, shown in Eq (7.39); the latter can exist in two energetically equal states, which is not the case for the in-channel geometry. If motion occurs by concerted jumps, diffusion can be described as a random walk of the center of mass, and follows the rules already laid out in Chapter 1. When dealing with the diffusion of single atoms, we derived the distribution of displacements in addition to the displacement fluctuation, as that provided information about the kinds of jumps participating in the diffusion. The displacement distribution has also been obtained for the diffusion of dimers on the same assumptions used here. From the distribution it is possible to arrive at the jump rates a and b, but knowledge of these is more readily attained from the displacement fluctuation combined with the ratio of occupation probabilities. For this reason we eschew discussion of the distribution of displacements and just refer to the literature [7].

526

Mechanism of cluster diffusion

Atom Position 0 1 2

COM

Cross-Channel

Transition Rate Configuration

0

a

1

b

2

In-Channel

1

a

2

b

3 Units Fig. 7.9

b

1

a

0

1 b

2

a

1

sec–1

0 /2

0

0

Comparison of atom motion in cross-channel and in-channel dimers, showing configurations and transition rates for motion by jumps of individual atoms [1].

aIII

aI

bI

cI aII

cI

bI

aI

bII cII bIII

cIII

A cII

bII cIII

bIII

aII

aIII B

x=

0

1

2

3

Fig. 7.10

Cross-channel trimer configurations and rate constants in diffusion by transitions of individual atoms [7]. Configurations of different energies at the same center of mass site x are distinguished by A and B.

7.1.2

One-dimensional movement of trimers The same general procedures can also be applied to the examination of larger clusters, such as one-dimensional trimer diffusion, but the schematic of trimer jump rates and configurations in Fig. 7.10 makes it clear that this is a more complicated task. This problem has been worked out by Wrigley et al. [7]. The details will not be considered here. The trimer can have a center of mass in the same place for two energetically different configurations, which are marked in Fig. 7.10 by A and B. The smallest displacement of the center of mass made by a trimer moving by individual jumps is ‘0 =3. The displacement fluctuation (in units of ‘0 =3) of the trimer center in the limit of long times is given by

527

7.1 Via single atom jumps



Δx2



¼ 4tP0A ½ðbI
cI þ bII
cII ÞΔA =Δ þ ðbIII
cIII ÞΔB =Δ þ cI ð1 þ aII =aI Þ þ ðbI þ 2bII þ 2cII Þ=2aI =cI

(7:41)

where ΔA ¼ ½bIII ðbII
aII cI =aI Þ
ðbI
cI þ bII
cII ÞðbIII þ cIII Þ

(7:42)

ΔB ¼ ½bII ðbI
cI þ bII
cII Þ
ðbII
aII cI =aI Þð2bI þ cI þ bII þ cII Þ

(7:43)

Δ ¼ ð2bI þ cI þ bII þ cII ÞðbIII þ cIII Þ
bII bIII

(7:44)

and P0A , the probability of finding a trimer in configuration 0A, regardless of the position x, is P0A ¼ ½1 þ 2aI =cI þ 2aII =cIII þ aI cII =ðcI aIII Þ
1 :

(7:45)

These tedious expressions simplify greatly under some obvious conditions. Should diffusion occur only through configurations A, as would be the case if aII ¼ bII ¼ cII ¼ 0, then 5ðΔxÞ2 4 ¼

18aI bI cI t : ð2aI þ cI Þð2bI þ cI Þ

(7:46)

If, on the other hand diffusion takes place through 0B, 1A, and 2A, as it would when bII ¼ cI ¼ 0, then the displacement fluctuation is given by 5ðΔxÞ2 4 ¼

18aIII bI cII t : ð2aIII þ cII Þð2bI þ cII Þ

(7:47)

Here it must be remembered that the unit displacement is always ‘0 =3.1 Cross-channel tungsten trimers were observed in the FIM by Graham and Ehrlich on a W(211) surface [8] and are shown in Fig. 7.11. All atoms in the trimer are resolved and it

Fig. 7.11

Tungsten cross-channel trimer diffusion on W(211) plane observed in FIM after diffusion at 277 K [8].

1

Also worked out has been the displacement fluctuation of dimers in one-dimensional diffusion undergoing dissociation, a rather more complicated problem [7].

528

Mechanism of cluster diffusion

Atom Position –1

COM

0 +1 +2 0

at

1

bt

ct

2A

2

ct

bt

2B

at

1

3 Units Fig. 7.12

Transition Rate Configuration

0/3

0

1

sec–1

Schematic of in-channel trimer motion, showing jump rates and spatial configurations assuming single atom transitions [1].

appears that, just as with cross-channel dimers, diffusion takes place by single atom transitions. Diffusion of in-channel trimers is easy to describe if it occurs by jumps of individual atoms. A schematic for such motion is given in Fig. 7.12, and by analogy with previous discussions of cross-channel trimers, the displacement fluctuation for the in-channel trimer is found as 

Δx2 ¼

2at bt ct ‘0 2 t : ð2at þ ct Þð2bt þ ct Þ

(7:48)

It must be emphasized again that everything has been based on the assumption that dimer and trimer atoms make individual jumps. This certainly seems to be the case for cross-channel clusters, but is probably not generally valid when all the cluster atoms lie in one channel, as in Fig. 7.9 or 7.12. It should be noted that we have furthermore assumed that clusters move by atoms making simple jumps, with no exchanges with the lattice. The latter jump mechanism would be readily discerned, as it would result in crosschannel transitions that are immediately apparent.

7.1.3

Leapfrog cluster diffusion So far we have considered diffusion in which a single cluster atom advances in some way; the rest of the cluster atoms follow one by one, and in this way the cluster moves over the crystal surface. An alternative mechanism to such diffusion was discovered by Linderoth et al. [9], who studied motion of platinum clusters on Pt(110)-(1 × 2). What they found is demonstrated by STM images as well as schematics in Fig. 7.13. Instead of the front atom of the in-channel tetramer displacing, the atom in the back jumps onto the top of the cluster, advances over it, and eventually returns to a position in front of what

529

7.1 Via single atom jumps

(a)

(c)

Edown

Eup

(b)

(d)

Fig. 7.13

Leapfrog movement of in-channel Pt tetramer on Pt(110)-(1 × 2) surface at 313 K. STM images at the left, schematics at the right. In (b) atom from end of cluster has transitioned to the top, and incorporated at the front in (c). (d) Schematic of leapfrog movement and potential for Pt end-atom diffusing over the cluster on Pt(110)-(1 × 2) (after Linderoth et al. [9]).

Fig. 7.14

Leapfrog movement of trimer. Left: Diffusion path of trimer. Right: Potential energy changes in leapfrog cluster motion (after Montalenti and Ferrando [10]).

had been the first atom. In this way the center of mass of the whole cluster changes its position. A schematic potential for this transition is given in Fig. 7.13d. It should be noted that such transitions are peculiar of the reconstructed fcc(110) surface, and are analogous to the metastable walks observed in single atom diffusion on Pt(110)-(1 × 2) and discussed in Section 3.3.3, in which the adatom can diffuse speedily by jumping up onto the sidewalls of the channel. Leapfrog diffusion over a cluster has been shown energetically advantageous in calculations with RGL potentials by Montalenti and Ferrando [10], whose results for an in-channel Pt3 cluster are shown in Fig. 7.14. In the absence of reconstruction of the fcc(110) surface, this process has so far never been observed and is unlikely to occur.

530

Mechanism of cluster diffusion

Fig. 7.15

Schematic of hetero-dimer states and rates of jumps occurring on bcc(110) plane [11]. (a) Via horizontal. (b) Vertical intermediates. (c) Grid for center of position COP, jump rates, and cluster configurations in diffusion via horizontal transitions on bcc(110) plane, assuming atoms make individual transitions. Solid lines indicate unit cell for COP grid.

7.1.4

Two-dimensional dimer movement Analysis has not been confined to just one-dimensional motion of clusters. The twodimensional movement of hetero-dimers has also been carried through [11], assuming that diffusion occurs by single atom jumps. On a bcc(110) surface, diffusion can occur via horizontal intermediates, shown in Fig. 7.15a, or through vertical intermediates, in Fig. 7.15b. Expressions have been worked out for the displacement fluctuation, again on the assumption that jumps of the two atoms take place independently. A grid of the positions of the dimer center, jump rates, and configurations is shown in Fig. 7.15c for motion via horizontal transitions, and the displacement fluctuation along the x-axis in the limit of long time intervals and in units of a‘ /4, where a‘ is the lattice spacing, turns out to be 5ðΔxÞ24 ¼

16bct : ½a þ c þ 2ðb þ dÞ

(7:49)

The pffiffiffimean-square displacement along the y-axis, again at long times and in units of 2a‘ =4, is given by

531

7.1 Via single atom jumps

5ðΔyÞ2 4 ¼

8act  8act=ða þ cÞ: ½ða þ cÞð1 þ a=2bÞ

(7:50)

When we take into account horizontal as well as vertical intermediate states, the displacement fluctuation along the x-axis changes to   ac eg 5ðΔxÞ2 4 ¼ 8P1 t þ ; (7:51) aþc eþg with the new rate constants defined in Fig. 7.15b. The displacement fluctuation depends on the time t, on the probability P1 of dimers being in an odd state, as well as on the effective rates of jumping out of that state. The probability P1 of dimers is given by P1 ¼ 1=ð1 þ

a e þ Þ: 2b 2f

(7:52)

For a simple dimer, with both atoms of the same material, c = a and d = b. For such a dimer the displacements along x and y are equal except for the length of the pffiffiffi unit step: a‘ =4 along x and 2a‘ =4 along y. The displacement fluctuation emerges as

  a2‘ aða
2bÞ 16 Δx2 ¼ 8 Δy2 ¼ 8abt þ  ½1
exp
2ða þ 2bÞt : (7:53) a þ 2b a þ 2b Atom jumps leading to dimer diffusion on the (100) plane are shown in Figs. 7.16a and b when simple hops take place. The same final configurations can be obtained in translation and rotation of dimers when atom exchange with the lattice occurs, as is described in the next section. From Fig. 7.16a and b it is clear that a COM displacement of a‘ /2 will occur along both coordinates when an atom moves by hopping. Rotation of the dimer as well as translation can be achieved by single atom hopping, however rotation requires only one jump while translation involves at least two. Based on the previous presentation it is possible to work out an expression for the mean-square displacement of a simple dimer on the (100) plane. After changing to and  axes at 45o to the Cartesian coordinates, as in Fig. 7.16c (which also defines the various symbols), this immediately gives the displacement fluctuation in Eq. (7.54)

 2  2 a2‘ aða
2bÞ Δ 8abt þ  ½1
exp
2ða þ 2bÞt : (7:54) ¼ Δ ¼ a þ 2b a þ 2b For long diffusion times this reduces to  Δ

2



 8abta2‘ : ¼ Δ2 ¼ a þ 2b

(7:55)

The mechanism by which rotation of the dimer takes place is not easy to establish, however. We have so far assumed that ordinary atom hops are involved in diffusion. An alternative is possible – diffusion by one of the atoms undergoing exchange with a lattice atom, as was illustrated in Fig. 3.53. This yields exactly the same displacement as ordinary atom jumps, and it may therefore not be simple to distinguish the two, even by measuring the distribution of displacements.

532

Mechanism of cluster diffusion

Fig. 7.16

Jump rates and configurations for dimer diffusion on a (100) surface [11]. (a) Simple dimer diffusion which results in translation of dimer. (b) Diffusion with rotation. (c) Schematic showing grid for center of mass as well as transitions for simple dimer on (100) referenced to and  coordinates, along which dimer states fall into either even or odd.

7.1.5

Cluster diffusion by atom exchange So far, the emphasis has generally been on simple atom jumps in the mobility of atom clusters over a surface. However, as was already pointed out in Chapter 3, exchange of an adatom with an atom from the substrate is known to play an important role on fcc planes. The possibility therefore also exists that such exchange processes may participate in the diffusion of clusters. For the sake of simplicity, we shall limit ourselves to the diffusion of dimers. A larger examination of the mechanisms of cluster diffusion has been given by Chang et al. [12], on which we have relied in this brief summary. The most obvious possibility of exchange holds for dimer motion on the fcc(110) plane. The usual picture for an in-channel dimer, shown in Fig. 7.17a, is for one atom to jump along the channel, with the second one following subsequently. Also possible, however, is a concerted transition of both atoms, as in Fig. 7.17b. Cross-channel motion of the dimer can also occur by single atom hopping from one channel to the next, but from what is known about single atom transitions this is probably a high energy process. A more likely possibility for a cross-channel transition is sketched in Fig. 7.18; one atom can carry out an exchange with an atom from the lattice, followed by a similar transition

7.1 Via single atom jumps

533

Fig. 7.17

Schematics of in-channel motion of dimer. (a) One dimer atom jumps, the second follows subsequently. (b) Two-atom movement of dimer.

Fig. 7.18

Cross-channel diffusion of dimer on fcc(110) by atom exchange, one atom at a time. (a) First dimer atom starting exchange with the wall atom. (b) First dimer atom creates dumbbell with wall atom, second atom starting exchange. (c) First atom finished exchange, while second atom creates dumbbell with the wall atom. (d) Both dimer atoms have finished exchange, creating dimer in next channel. Arrows show all possible directions for exchange, black arrows indicate direction chosen for illustration. Dumbbell transition state is marked by ellipse.

by the second atom. We illustrate one possible way for the dimer to exchange in a singleatom event. Arrows in Fig. 7.18a indicate possible directions of exchange, the black arrow shows the direction chosen for the illustration. The intermediate stage of the exchange mechanism is creation of a dumbbell, as already discussed in Chapter 3 and shown in Fig. 3.42. Dissociation of the dumbbell can proceed in four directions on an fcc(110) surface. For single atom exchange all these directions are supposed to be equal; however, experiments indicate otherwise. In the case of a dimer after creation of the dumbbell, shown in Fig. 7.18b, it is clear that dissociation of the dumbbell will not occur with the same probability in all indicated directions, since dissociation will be influenced by the presence of the second adatom on the surface. Exchange of only one dimer atom will result in the creation of a cross-channel dimer. As an alternative, the two dimer atoms

534

Mechanism of cluster diffusion

Fig. 7.19

Cross-channel diffusion of dimer on fcc(110) by a two atom exchange process. Dumbbell transition state is marked by ellipse.

Fig. 7.20

Dimer diffusion on (100) plane. (a)–(c) By single atom exchange with substrate atom. (d)–(f) By two atom exchange with surface atom. Arrows indicate all possible directions for exchange, black arrow indicate direction chosen for illustration. Dumbbell transition state is marked by ellipse.

may also undergo simultaneous exchange, for which one possible path is illustrated in Fig. 7.19. We suspect that simultaneous exchange with atoms moving in two different directions and condensing after exchange is rather unlikely. Distinguishing between atom-by-atom and simultaneous exchange processes experimentally is not straightforward, but calculations of the activation energies for the two processes should give us some insight into what actually happens. Single atom exchange processes have not been confined to fcc(110) planes –these have also been detected on (100) surfaces, so it is likely that diffusion of dimers may occur in this way. A single atom exchange process for a dimer is depicted in Fig. 7.20a–c, resulting in dimer rotation. Exchange proceeds in analogy to the process on fcc(110): the first dimer atom can exchange with a surface atom in four possible directions, indicated in Fig. 7.20a by arrows; the black arrow shows the direction we chose for the illustration. Not all directions will be equal due to the presence of the second atom on the surface. The directions away from the dimer are likely to be less probable. The intermediate stage of the exchange will also be a dumbbell; however, on this surface there are only three possible directions for dumbbell dissociation, as shown in Fig. 7.20b, and one

535

7.1 Via single atom jumps

of them (out from the other atom) is less likely than others. A translation of the dimer can be obtained by double rotation or without rotation. As shown in Fig. 7.20d–f, both atoms may execute a simultaneous exchange. On the (100) plane, the occurrence of dimer rotation provides a simple criterion to distinguish exchange, atom-by-atom, from simultaneous exchange of both atoms, provided it is known in an experiment that only exchange occurs. However, the same configurations can be achieved by a sequence of hops on the surface. The coexistence of movement both by hopping and exchange has been detected in experiments of atom diffusion. Such coexistence might be also occur in dimer movement, complicating the interpretation of cluster diffusion.

7.1.6

Peripheral diffusion and the evaporation–condensation mechanism In diffusion of larger clusters, movement by single atoms is usually associated with a sequence of peripheral displacements. Here it must be noted that the rate of such perimeter movement depends on the kind of cluster edge at which the atom is moving. Rates are also different for crossing corners (known as corner rounding) or for attaching– detaching from kinks; the last one depends on the number of kinks present. As an example, for the cluster on an fcc(111) plane shown in Fig 7.21a, movement along steps A proceeds at a rate as, while along steps B it occurs at a rate bs; movement around corners takes place at the rate hr . Corner breaking at the rate hre, in which an atom detaches from the corner and starts movement along the cluster edges, might also play a role. When in addition kinks are present, the rates of detachment hk from a kink to the straight edge and attachment kk from a straight edge to kinks, shown in Fig. 7.21b, also

hc

hr ka

as

ka

hse

bs kk

ha

kke kke

hk

(a) hc

hke

(c) dt dt

hse

ka ka kke

(b1) Fig. 7.21

ka

dt

ha hke

(b2)

dt

(b3)

Kinetic processes in peripheral displacements. (a) Perimeter movement. (b) Correlated evaporation–condensation. (c) Evaporation–condensation mechanism.

536

Mechanism of cluster diffusion

have to be taken into account. For some structures, movement along the edges might proceed by the exchange mechanism; for example, for Al(111), theory predicts the exchange mechanism at B-type steps as a leading contender [13]. Movement of the center of mass will be described by a combination of all these rates together, and the problem is further described in Section 7.5. At higher temperatures, diffusion along the steps can proceed by longer jumps as well, which will proceed at different rates. In the movement of clusters by correlated evaporation–condensation (CEC), known also as the terrace limited mechanism, cluster atoms separate from the cluster and after diffusion on the terrace come back to the same cluster and combine again. In such mechanism at least four rates are involved for a compact cluster. The first one indicated in Fig. 7.21b as hse is the rate of detachment of an atom from the straight step or at the rate hc from the corner (core breakup) to the terrace; after detachment the adatom moves on the terrace with a rate typical for terrace diffusion, indicated in the picture as dt. Finally, when the atom meets the edge of the cluster it can again attach, at the rate of ka if the adatom meets a straight edge. When kinks are present, different rates hke will be associated with detachment of atoms from the kink to the terrace and kke for attachment of adatoms to the kink from the terrace. When a compact cluster has an atom attached to a straight step, this atom usually detaches first at the rate of ha. In carrying out simulations of diffusion by these mechanisms, all the different rate constants must be included to attain a proper description – a single rate will not do justice to this problem. However, diffusion around cluster edges is expected to take place over smaller barriers than atom detachment, so that at low temperatures the latter process should not be important. One can also expect that attachment processes are very rapid and influence movement less than detachment. In the last named mechanism, evaporation–condensation (EC), shown in Fig. 7.21c, the cluster atom leaves the cluster for the terrace, but a different adatom from the terrace attaches to the cluster, maintaining constant average cluster size. The rates for such a mechanism are the same as in the correlated evaporation–condensation mechanism, but the existence of such a mechanism will depend on the presence of adatoms on the terrace at some distance from the cluster. Both EC type mechanisms are rather unlikely at low temperatures, due to the high energy of detachment.

7.2

Concerted displacements The examples of cluster motion considered up to this point have all depended upon jumps of individual atoms. This may not always be the case, however, and in this section we will present examples of cluster diffusion by concerted motions.

7.2.1

Gliding A good demonstration of concerted atom diffusion is provided by the diffusion of hexagonal clusters made up of 7 or 19 atoms on an fcc(111) surface, in the literature sometimes described as magic clusters. For Ir(111) such diffusion has been examined

7.2 Concerted displacements

537

Fig. 7.22

Motion of compact hexagonal Ir19 cluster over Ir(111) plane after heating to ~ 690 K [14]. FIM images on top, schematics below. No individual atom transitions were detected.

Fig. 7.23

Pattern of Ir7 displacements on Ir(111) obtained from Sutton–Chen potential, for vibrational mode of 0.75 THz [16].

by Wang [14,15], and is shown for an hexagonal Ir19 cluster in Fig. 7.22. What is clear here is that the center of the cluster moves over the surface, but there is no evidence in the FIM images of any shape changes, suggesting that the cluster glides over the surface as a whole. Atomic simulations to explain details of this diffusion for Ir7 have been carried out by Kürpick et al. [16] using Sutton–Chen potentials. Shown in Fig. 7.23 is a view of the pattern of surface atom displacements in a vibrational mode at 0.75 THz during movement of an Ir7 cluster. What is very surprising in this study is that the surface is not rigid and seems to create an easy path for moving the cluster. A more detailed picture was obtained in molecular dynamics simulations at the elevated temperature of 1350 K. In Fig. 7.24 are given three separate sequences, indicating distortions of the atomic arrangement as the cluster moves from one equilibrium site to a neighboring one. However, on reaching equilibrium, the hexagonal shape

538

Mechanism of cluster diffusion

Fig. 7.24

Molecular dynamics simulations with Sutton–Chen potential for Ir7 diffusing on Ir(111) at 1350 K [16]. Black circles show initial sites of minimum energy. Cluster undergoes distortions while maintaining overall shape.

3

4 E2b

2

1

3

4

3

4

2

1 3

4

Ec 2 Fig. 7.25

1

2

1

Competing mechanisms for tetramer motion on (100) plane [17]. Top: by severing nearestneighbor links. Bottom: by shearing of dimer.

is always recovered. It is evident from this study that this hexagonal cluster does not diffuse over the surface by jumps of individual atoms, and that concerted motion of the whole cluster is involved.

7.2.2

Cluster shearing and reptation In 1996, Shi et al. [17] came out with a novel way for a cluster to diffuse on a (100) surface – by shearing rather than by simple atom jumps. The two types of processes are illustrated in Fig. 7.25 for a tetramer: movement by a single atom, and by two atoms together in a shearing motion. Is shearing an energetically favorable process? To examine this question, Shi et al. did EAM calculations for the two events in three systems, as indicated

539

7.2 Concerted displacements

Table 7.1 Self-diffusion of clusters on (100) surfaces by shearing, with barriers in eV [17]. Second barrier is for cluster atom detachment. Cluster Geometry

Cu(100)

Ag(100)

Ni(100)

0.503

0.478

0.632

0.494

0.480

0.611

0.501/0.552

0.491/0.503

0.621/0.673

0.688/0.815

0.637/0.723

0.842/0.989

0.551

0.510

0.675

0.713/0.835

0.658/0.737

0.870/1.008

0.552

0.512

0.677

0.758/0.838

0.676/0.740

0.918/1.011

0.857/0.975

0.763/0.903

1.051/1.215

0.554

0.516

0.678

in Table 7.1; for clusters of 4, 6, or 8 atoms they found dimer shearing rather than single jumps to be the low-energy process. The shearing process also can explain the oscillatory behavior of cluster motion observed in experiments by Kellogg [18] for self-diffusion on the Rh(100) surface. Depending on the size of the cluster and the surface geometry, dimer or trimer shearing may be energetically favorable. Three years later, a related mechanism on fcc (111) planes, reptation, was put forth by Chirita et al. [19]. In this process, part of a cluster glides over the surface, and is then followed by the remaining atoms of the cluster, as illustrated in Fig. 7.26. The two parts need not be of the same size. Movement proceeds by a high-energy transient state, in which part of the cluster is in fcc sites and a second part in hcp sites. The cluster can relax back from this state or move forward to a new position. Estimates of the diffusion barrier were made with EAM potentials and for clusters with more than seven atoms, reptation was found to be the low-energy process.

7.2.3

Dislocation mechanism Hamilton et al. [20] in 1995 made an interesting suggestion about diffusion of clusters on fcc(111) surfaces – it could occur by nucleation of a dislocation. For motion of nickel

540

Mechanism of cluster diffusion

(a)

t = 0 ps (b)

t = 1 ps (c)

1 2 3

1 2 3 4 5 6

4

5 6

t = 2 ps

1 2

3 4

5 6

-[112]

[110] Fig. 7.26

Schematic for diffusion of hexamer on fcc(111) surface by reptation. Initial positions indicated by black circles. Bottom shows subcluster units (after Chirita et al. [19]).

Fig. 7.27

Motion of 19 atom cluster over fcc(111) surface via a misfit dislocation. Arrows show motion toward bridge positions, allowing cluster to diffuse (after Hamilton et al. [20]).

clusters on the Ni(111) plane they made calculations for islands of 19, 27, 37, and 43 atoms using EAM potentials. Motion for a cluster of 19 atoms via a dislocation is illustrated in Fig. 7.27. They came to the conclusion that for islands in the range of 20 to 100 atoms, dislocation motion by gliding of cluster rows should be the process of the lowest activation energy. The activation energy to create a misfit on a Ni(111) surface was calculated as 0.268 eV per atom. However, for clusters bigger than 100, they estimated that kink motion has the lowest energy requirement, making it the more likely mechanism of movement.

541

7.2 Concerted displacements

(d)

V(x)

0.1 0.2 0.3 100 meV

0.4 0.5

120

Activation Energy ED(meV)

0.0

Dislocation Position in Units of Island Length

(a)

100 80 60 40 20 0 0.87

0.88

(e)

0.89 Misfit

0.90

1000 x

100

10 0

0 0.2 0.4 0.6 0.8 1 Dislocation Position in Units of Island Length

Fig. 7.28

Jump/ns

20

10 61 Atom Ag Island 1

300 200 0.1

100 0

2

2.5

(f) 6

7

8 9 10 11 12 Island size

Activation Energy ED(meV)

(c)

Ag monomer

30

Activation Energy ED(meV)

Island Energy (meV) (relative to minimum)

(b)

3.0

3.5

4.0

4.5

5.0

450

500

1000/T 230 220 210 200 190 180 200

250

300 350 400 Temperature T(K)

Cluster motion via nucleation of a dislocation. (a) Position of atoms in cluster of nine in onedimensional diffusion over substrate potential is shown on top. (b) Energy of island relative to minimum in its dependence on position of dislocation. (c) Activation energy for diffusion via dislocation mechanism as a function of island size. (d) Activation energy for diffusion via dislocation mechanism as a function of misfit. (e)–(f) Diffusion of silver islands calculated for Ru(0001) plane. (e) Plot for single silver adatom and for island of 61 atoms, the latter fit with activation energy for diffusion of 61 atom Ag cluster varying with temperature (after Hamilton [21]).

Hamilton [21] also examined diffusion of heteroepitaxial islands, in which the lattice constant of the island differed from that of the underlying crystal. Calculations were made on the Frenkel–Kontorova model [22] using EAM techniques for a cluster with a spacing 11% smaller than that of the substrate. In Fig. 7.28a is shown a one-dimensional island of nine atoms in the potential of the substrate. A dislocation was nucleated at the 0.1 position on the left side of the island; the 0.0 schematic shows the dislocation-free island. In schematic 0.2, a dislocation starts to move and at 0.5 is in the center of the

542

Mechanism of cluster diffusion

island. Introduction of a dislocation into the cluster changed the energy as shown in Fig. 7.28b, resulting in an activation energy of only 0.025 eV, compared to 0.10 eV for diffusion of a single adatom in the same potential. As the size of the island changed so did the activation energy, as indicated in Fig. 7.28c, reaching a minimum value for an island of nine atoms. The diffusion barrier is of course a function of the misfit between the island and the crystal, shown in Fig. 7.28d, which is dependent upon the temperature. Hamilton also calculated the energetics of two-dimensional islands of silver on Ru(0001); the activation energy for diffusion was found to be a sensitive function of the temperature. This is indicated in Fig. 7.28e–f for an island of 61 silver atoms. What is clear is that the movement of the 61 atom island cannot be fitted by single-atom diffusion; the authors attribute this to the temperature dependence of the misfit between adlayer and substrate. The paper showed the dependence of the activation energy for diffusion on the misfit as well as the size of the clusters. These are very interesting ideas, but right now only little information is available about the behavior of hetero-epitaxial islands, so the importance of these notions is difficult to judge.

7.2.4

Concerted translation and rotation on FCC(111) Hamilton et al. [23] have also come up with another intriguing mechanism for cluster diffusion. On putting a layer of iridium on an Ir(111) plane, they discovered in their firstprinciples calculations that such a pseudomorphic layer had a minimum energy configuration when the cluster atoms were on top of the substrate atoms slightly above a stable fcc configuration. With this as a start, they conjured up the four mechanisms in Fig. 7.29 in which concerted translation or translation plus rotation is accomplished for hexagonal clusters. The activation energies found for the different mechanisms are shown in Table 7.2, and it is clear that bridge glide offers the lowest energy path. However, cartwheel shuffle and cartwheel glide have a not much higher energy for 7 atom clusters and might participate at higher temperature. For 19 atom clusters the energy of a cartwheel shuffle is even closer, making coexistence possible. Whether such a pathway exists for other clusters as well still remains to be explored.

7.2.5

Concerted exchange mechanisms Molecular dynamics simulations of dimer diffusion have been carried out by Zhuang et al. [24] on the (100) surfaces of platinum, silver, aluminum, and gold using EAM potentials. For platinum and silver, interactions were from the EAM potentials of Haftel et al. [25–27], for aluminum and gold from Johnson [28]. The mechanism for dimer diffusion preferred on all these surfaces is the conventional exchange process, shown in Fig. 7.20, or simple hopping. It should be noted, however, that on Pt(100) the hopping event does not occur. However, other processes, which take place less often, were also observed in this study. These are interesting nevertheless and are shown in Fig. 7.30 for a temperature of 700 K. In the 180o exchange rotation, adatom 1 interacts with lattice atom 2, lifting it out of the surface. This atom then interacts with atom 3, also raising it. Atom 1

543

7.2 Concerted displacements

Table 7.2 Energetics (in eV) of concerted motion of iridium clusters on Ir(111) [23].

Fig. 7.29

Cluster size

Bridge glide

Top glide

Cartwheel shuffle

Cartwheel glide

7 atoms 19 atoms

1.54 3.2

3.02 5.2

1.92 3.6

1.92 5.1

Possible movements of 19 atom Ir cluster on Ir(111) surface by concerted atom displacements. White atoms serve to indicate rotation. Energies are indicated in Table 8.2 (after Hamilton et al. [23]).

then sinks back, as does atom 2, leaving a dimer rotated 180o with respect to the original position. The trajectory in Fig. 7.30f illustrates this process more clearly. Also observed on Pt(100) was a 270o exchange rotation, in which the final configuration of the dimer is 270o rotated compared to the original one. For Al(100), simulations at 450 K revealed both 180o and 270o exchange rotations. Also possible is a 360o exchange rotation, rarely observed for Pt(100) and Al(100). The authors labeled these for Pt(100) and Ag(100) as concerted complicated exchange events, and for Al(100) as strain-induced events. For platinum at 950 K exchange rotation events amounted to roughly 10% of conventional exchange, for aluminum at 550 K to only 6%. A diagram of the energy changes in 180o exchange rotation, which is observed most frequently, is given in Fig. 7.31, keyed to the pictures in Fig. 7.30. Other processes, such as simultaneous jumps of both atoms, and others, were also identified in the simulations, but only quite rarely. Liu et al. [29] extended calculations by considering in addition Cu(100) as well as Ni(100), and relying on three different potentials – EAM interactions [28],

544

Mechanism of cluster diffusion

4 1

0

1

3

2

4

4

0

1

3

2

(c)

4 0 1

4 0

3

3

1

2

(d)

2 (f)

(e)

4

4

3 3

0 1

3

2

(b)

(a)

0

0 2

1

2

(g)

(i)

(h)

4 1

0

4 0

3

2 (j)

3 2

1

(l)

(k) 4

4 3

0 1

0 1

2

3 2 (o)

(m) Fig. 7.30

(n)

Exchange events during dimer diffusion on fcc(100) plane. (a)→(b)→(c)→(d)→(e) 180o exchange rotation. (a)→(b)→(c)→(d)→(g)→(h) 270° exchange rotation on Pt(100) at 700 K. (f), (i) Molecular dynamics trajectories. (a)→(j)→(k) and (a)→(m)→(n) depict 180° and 270o exchange rotation on Al(100) at 450 K. (l) and (o) give molecular dynamics trajectories (after Zhuang et al. [24]).

surface-embedded-atom potentials (SEAM) [25–27], and RGL interactions [30,31]. The results for static activation energies are given in Table 7.3 for dimer hopping as well as simple exchange on (100) surfaces. For nickel and copper, hopping occurs over the lower barrier, but for aluminum, gold, and platinum, the situation is reversed and hopping has to proceed over a higher barrier. The first group, for which Ee > Eh, is classified as hard, the second, where Eh > Ee, as soft. For silver, different potentials give inconsistent results, as shown in Table 7.3, making classification of this material difficult. For platinum the dimer in the transition state is actually locally stable, and this is therefore referred to as a soft surface.

545

7.2 Concerted displacements

Table 7.3 Activation energy of hopping and simple exchange mechanism on metal(100) surfaces [29]. Activation energy Surfaces

Hopping (eV)

Simple exchange (eV)

Ag(E) Ag(R) Ni(E) Cu(R) Al(E) Ag(S) Au(E) Pt(E) Au(R) Pt(S)

0.4836 0.5001 0.5607 0.4307 0.4614 0.5818 0.8151 1.1976 0.5962 1.2319

0.5892 1.0112 1.349 0.7853 0.3545 0.3615 0.2647 0.6437 0.3715 0.2923

E-EAM, R-RGL, S-SEAM potential 0.5

single adatom dimer

Energy E(eV)

0.4 0.3 0.2 0.1 0

Fig. 7.31

(a)

(b)

(c) (d) Structure

(e)

Variation of the energy during exchange rotation of the dimer on Pt(100). Solid curve gives changes for 180o exchange rotation. Letters refer to the structures in Fig. 7.30 (after Zhuang et al. [24]).

In addition to simple exchange and exchange rotation, already discussed above, other diffusion mechanisms were also found in molecular dynamics simulations. These were cooperative hopping, hopping rotation, and cooperative exchange. In cooperative hopping, illustrated in Fig. 7.32, atom A of the dimer hops over the potential barrier, at the same time this movement triggers an instantaneous jump of atom B to a nearest-neighbor site. The directions for the jump of atom B are fixed by the movement of atom A; atom B can jump only towards atom A horizontally (c2) or vertically (c1). Hopping rotation is made up of two consistent hopping events which rotate the dimer, shown in Fig. 7.16b; cooperative exchange occurs via a two-atom dimer exchange, already shown in Fig. 7.20d–f. These three movements occur not too frequently by comparison with simple hopping and exchange. It must be emphasized that on middle surfaces, exchange of the strain-induced kind predominates; on soft surfaces, concerted motion exchange is the important process.

546

Mechanism of cluster diffusion

A B A B (a)

A

(c1)

B B

(b)

A

(c2) Fig. 7.32

Cooperative hopping of dimer on Cu(100) surface, obtained in RGL calculations based on the work of Liu et al. [29].

7.3

Mechanism of dimer diffusion versus bond length A simple model of dimer diffusion in a two-dimensional periodic potential has been given by Pijper and Fasolino [32]. The two atoms in the dimer at a separation r interact via a Morse potential, and the resulting potential V(s) for diffusive motion is shown in Fig. 7.33a, where a‘ is the substrate lattice constant. Movement of a rigid dimer from position A1 to the neighboring site A2 will occur via saddle point T3 when the dimer moves as a unit, with r = req. Piecewise diffusion, in which stretching of the dimer is possible, occurs over the path A1 → T1 → B → T2 → A2, where B is a local minimum. During such a movement the dimer bond length almost doubles after the first atom jump and regains its length after diffusion. This piecewise rotation may be favorable for a larger width w of the Morse potential. Pijper and Fasolino looked at dimer movement as a function of stiffness. For larger width w (smaller force constant) piecewise motion became energetically more favorable, as is apparent in Fig 7.33a. A similar effect occurs changing the effectiveness of DM/V0, where DM is the depth of the Morse potential and V0 that for the atom-substrate energy, as is shown in Fig. 7.33b. The broken lines indicate piecewise movement, solid lines rigid movement and the ratio DM/V0 is proportional to the force constant kforce. An increase in DM/V0 increases the barrier and makes piecewise diffusion less favorable than rigid diffusion. Additionally to stretching of dimers the authors also looked at movement of dimers with a contracted bond (r < req). They found that movement of such a dimer should proceed over a lower energy barrier than for a rigid or a stretched dimer, as indicated in Fig. 7.33c. The model suggests that movement of dimers over a surface is the result of gaining energy due to interactions between the dimer atoms and losing of energy due to stretching the dimer bond during movement. Further investigations will be required to see if this model describes the nature of dimer movement correctly on real surfaces. It was already shown that using Morse potentials was rather unsuccessful in predicting the correct values of activation energies for single atom movement; however, the trends might be better described by this potential.

547

7.4 Kinetic mechanisms of larger clusters

1.5

1.5 DM /V0 = T2

er

T 0.5

w = 2.25a

0.2

(b) 0.4

0.6

0.8

distance s/stot 1.5

0.0 0.0

1.0 A2

T 1.0

B 0.5

rig id

0.0 0.0 A1

2.0

1.0

rig id

0.5 (a)

T3 B

di m er

T1

w = 0.75a V(s)/V0

1.0

di m

V(s)/V0

w = 1.5a

0.2

0.4 0.6 0.8 distance s/stot

1.0

cpd-correlated piecewise diffusion rd-rigid diffusion pd-pinched diffusion

cpd

V(s)/V0

1.0 rd pd 0.5

(c)

0.0 0.0

0.2

0.4 0.6 distance s/stot

0.8

1.0

Fig. 7.33

Energy changes for movement of dimer between adsorption sites. (a) For different values of the width w of the Morse potential. (b) For different values of DM/V0 where V0 is the effective barrier for non-interacting atoms and DM is the well depth between two atoms. (c) Comparison of extended, rigid and contracted dimer bond (after Pijper and Fasolino [32]).

7.4

Kinetic mechanisms of larger clusters The movement of large clusters has been found to have a significant influence on diffusion over surfaces, so a number of investigations has been carried out to get insight into such diffusivity. We will explore this problem in somewhat more detail in the next section. It is interesting to note that in 1994 the center of mass of large clusters on an Ag(100) surface was found to proceed by a random walk on the surface [33], as shown in Fig. 7.34. However, the mechanism of diffusion constitutes a much more complicated problem than the movement of the center of mass. For big clusters the most probable mechanism is peripheral movement of cluster atoms, shown in Fig. 7.21a. Needless to say, the expressions for the diffusivity become complicated as cluster size increases, and have so far not been applied to real systems. For this reason we will not examine the derivations in any detail. Investigations of diffusion for bigger clusters started with the work of Voter [34] in 1986, who looked at Rh clusters on the Rh(100) surface with molecular dynamic and lattice gas simulations, relying on Lennard-Jones interactions. He concentrated on movement by sequential jumps and found that jumps away from a nearest-neighbor atom, as in

548

Mechanism of cluster diffusion

900

s

s

f

y(Å)

1800 f 800 1700 (a) –1250

(b) –1200 x(Å)

–1150

–1000

–950

–900

x(Å)

Fig. 7.34

Trajectories for two Ag clusters of different size on Ag(100) measured with STM. Start is indicated by s, and final location by f. (a) Cluster of 100 atoms. (b) 290 atoms (after Wen et al. [33]).

Fig. 7.35

Schematics for atomic jumps of rhodium atoms and their activation energies calculated with Lennard-Jones potentials relevant for diffusion of rhodium cluster on Rh(100) [34].

Fig. 7.35b, required more energy than for a single atom jump on the surface, as in Fig. 7.35a, demonstrating that cluster interactions are attractive. Moving along an edge, in Fig. 7.35f, was easier than jumping out of a block of atoms, shown in Fig 7.35d. Considerable progress has been made recently by Sanchez and Evans [35] who looked at self-diffusion of clusters, taking into account different hopping rates in movement on the square fcc(100) surface. Their investigation did not allow for the possibility of an exchange mechanism on this surface, or for atom detachment and attachment. However, they considered a number of different perimeter events, such as straight edge hopping, which proceeded at the rate he, corner rounding at a rate hr, kink escape at rate hk, and core breakup at the rate hc. The first two transitions did not require changes in the bonding or in the energy of the system, while the second two required breaking of one bond. They also considered the edge “breakup,” which involves breaking two bonds. For small clusters, such transitions are in the negligible range, but play a role for bigger islands.

7.4 Kinetic mechanisms of larger clusters

549

A number of other transitions should probably be considered in explaining the movement of real systems, for example diffusion of dimers around the perimeter, movement by longer jumps, as well as kink movement; however, as a starting point, the rates presented are of interest. Sanchez and Evans carried out detailed investigations for smaller clusters first; then they turned their attention to bigger entities. We will not present here their work for small clusters, as this was described in previous sections; however, we will present a few comments. For dimers, the authors did not consider the possibility of clusters being spread out over a distance longer than a nearest-neighbor separation. With this assumption, the movement of a dimer can only take place by transition between vertical and horizontal states, and the possibility of a dimer being in an intermediate state was ignored. This assumption greatly simplifies their calculations, since instead of the two rates of jumping a and b, that describe the conversion of a vertical into a horizontal dimer, or the reverse, shown in Fig. 7.16, they have only one, hr. The same assumptions were applied in the investigation of trimers, which led them to the diffusivity D¼

hr he ; 3ðhr þ he Þ

(7:56)

where he indicates the rate of straight edge hopping. For tetramers the diffusivity was described by D¼

6hr hc ; ðhr þ 18hc Þ

(7:57)

with hc being the rate of core breakup. The assumption that cluster atoms are always in nearest-neighbor positions and that the detachment–attachment mechanism is not active may describe the movement of big clusters quite correctly. Sanchez and Evans [35] arrived at relatively simple expressions for the mean-square displacement and diffusivities of larger clusters on (100) surfaces by refining the work of Titulaer and Deutch [6]. We will here illustrate an example for pentamers. A low energy path for diffusion of such a cluster is illustrated in Fig. 7.36.

Fig. 7.36

A low-energy pathway for self-diffusion of pentamer on fcc(100) plane [35].

550

Mechanism of cluster diffusion

Such a path is chosen to avoid direct core breakup, and movement is mediated by corner rounding or kink escape, whichever is more efficient. The assumption is made that the rates hr and hk remain finite, and Sanchez and Evans considered the limit he → 1, which, from detailed balance, requires that hc → 0. If unfolding of the configuration shaped like a C is ignored this leaves only the rates hk and hr. The diffusivity D for the pentamer ends up quite simply as D5 ¼

hr hk : 8ðhr þ hk Þ

(7:58)

Sanchez and Evans also briefly looked at tetramer diffusion with dimer shearing active, and later gave expressions for other clusters, which, however, will be downplayed here to concentrate on the work of Salo et al. [36] that expanded considerations of the diffusivity of clusters on (100) surfaces and considered both dimer and trimer shearing. Salo et al. looked at cluster made up of 5 to 9 atoms and followed the same nomenclature as that of Sanchez and Evans [35]. Concentrating first on pentamers as our primary example, we display in Fig. 7.37 the different rate processes assumed as being involved in diffusion of the clusters. The activation energies for these events have been evaluated by Trushin et al. [4] for copper on Cu(100) using effective medium theory, and this makes it possible to sort out transitions, indicated by italics in Fig. 7.37, that occur either very rapidly or slowly and can therefore be neglected; the rate limiting step is shown bold-faced. The remaining configurations for pentamer diffusion are indicated in Fig. 7.38 and yield the diffusivity D5 ¼

2½h0si þ 2ðh0s þ hr Þ ½2hsi ðhs1 þ hsi Þ þ hr ðhs1 þ 2hsi Þ ; ½2ðhs1 þ hsi Þ þ hr  ð8h0si þ 16h0s þ 21hsi þ 16hr Þ

(7:59)

where hs indicates the rate of shearing and h0s the reverse process. If hs1 ¼ hk =2 and other rates for dimer shearing are set to zero, Eq. (7.58), the result previously obtained by Sanchez and Evans [35], is found. Expressions for other clusters, containing from 6 to 9 atoms, were also derived. Using the rate constants for jumps of copper on Cu(100) from Trushin et al. [4], the diffusivities at different temperatures were calculated and are shown in Fig. 7.39. What is clear are the oscillations with size, with local minima for clusters of 4, 6, and 9 atoms, already seen in prior studies. New processes of dimer and trimer shearing (described earlier) increase the diffusion rate and affect island diffusion. What is evident is that knowing the individual jump rates it is now possible to write out the cluster diffusivities, at least on (100) surfaces. Finding these rates, however, still remains a task to be addressed in experiments.

7.5

Derivation of the mechanism of large cluster movements Observations of the movement of large clusters triggered a number of discussions of how the cluster diffusivity D was affected by the size and the specific mechanism of diffusion. Soler [37], in 1994, did Monte Carlo calculations with a triangular lattice model. In his

7.5 Derivation of the mechanism of large cluster movements

551

(a) N = 5

he = 0.21 hs1 = 0.42 hk = 0.51 hsi = 0.53 hr = 0.55 hs = 0.71

h3 = 0.82

hk' = 0.15 hsi' = 0.20 hs' = 0.39

h- = 0.65

(a) N = 6

hsi = 0.70

hs = 0.71

h- = 0.27

h- = 0.39

hc = 0.83

h3 = 0.90

hs' = 0.41

hsi' = 0.08

h- = 0.44

hk' = 0.15

hr = 0.55

h- = 0.71

(c) N = 7

hs1 = 0.45

hc = 0.84

hsi = 0.43

hs1' = 0.44

h3i = 0.50

hr = 0.55

(d) N = 8

hs1 = 0.45

hc = 0.84

hk = 0.54

h3 = 0.94

h3i = 0.63

h3i' = 0.31

hs = 0.76

h3' = 0.62

(e) N = 9

h3i = 0.81

h3a = 0.37

Fig. 7.37

hc = 0.82

h3 = 0.95

hs1 = 0.46

hk = 0.54

h3i' = 0.18

h- = 0.62

hk' = 0.18

h3' = 0.64

Jump processes for islands in which the number of atoms ranges from five to nine atoms on fcc (100) plane (after Salo et al. [36]). Activation energies for jump processes of Cu islands on Cu (100), evaluated by EAM [4]. Italics show processes not included in transition matrix as either too fast or slow. Rate limiting step is shown in boldface; hs describes a shearing process, hs′ reverse shearing process.

552

Mechanism of cluster diffusion

Fig. 7.38

Quasi-configurations for pentamer diffusion on fcc(100) plane, with multipliers giving the degeneracy (after Salo et al. [36]).

108

Diffusivity D

106 104 102 100 10–2

0

1

2

3

4

5

6

7

8

9

10

Island size Fig. 7.39

Copper cluster diffusivity (units a2‘ /sec) on Cu(100) as a function of size at temperatures of 300 (bottom), 400, and 500 K. Solid line –Monte Carlo simulations, open circle –analytical results with single atom processes, filled square – many-particle processes included (after Salo et al. [36]).

investigations he considered motion of large clusters by (a) movement along the perimeter, (b) movement via an evaporation–condensation mechanism, and (c) via vacancy movement. In Fig. 7.40a, the Arrhenius plot for clusters consisting of 300 atoms is presented. We can see that movement along the perimeter has a lower energy than an evaporation–condensation process or vacancy movement; the two latter mechanisms required comparable energy. The dependence of diffusivity on the clusters size, for all three mechanisms, is shown in Fig. 7.40b. Soler suggested that perimeter diffusion of atoms was the significant mechanism, evaporation–condensation being negligible at low temperatures. Van Siclen [38] then considered the role of correlations in the evaporation–condensation mechanism of diffusion. For uncorrelated events, the diffusivity D varied as 1/Rr, where Rr denotes the cluster radius. However, an evaporating atom is very likely to redeposit close to the point of emission, and this changed the dependence to D / 1/Rr2. In case of cluster diffusion caused by an atom running along the perimeter, the dependence of the diffusivity was given by D / 1/Rr3. His investigations viewed the cluster as circular and the structure of the substrate was not taken into account. He also investigated facetted islands and concluded that such islands could only exist above a certain size; for

553

7.5 Derivation of the mechanism of large cluster movements

Diffusivity

Diffusivity

10–6

10–9

10–6

10–12 10–9 (b)

(a) 10–15 2

4

6

8

10

εAA /kT Fig. 7.40

10

100 Cluster size

1000

Cluster diffusivities calculated for a triangular lattice from MC simulations. (a) Arrhenius plots of 300 atom cluster (circles). Contribution of evaporation–condensation processes is indicated by squares, of vacancy movement inside cluster by triangles. (b) Size dependence of cluster diffusion, separated as in (a) (after Soler [37]).

such clusters the diffusivity would be independent of the size. A year later, Soler [39] examined diffusion of clusters by evaporation and condensation of atoms as affected by correlations, and came to the conclusion that correlation depended on the diffusivity as well as on cluster size. He claimed that the dependence of diffusivity on 1/Rr was valid only for short times due to the increasing importance of the correlated evaporation– condensation mechanism. Under experimental conditions the diffusivity varied with size according to D  Rr
ð2þΔbÞ ;

(7:60)

where Δb describes the non-negligible perimeter energy, which was positive for atom clusters and negative for vacancy clusters. Diffusion of large clusters was analyzed in a rather different way by Khare et al. [40,41], who pointed out the analogy with equilibrium fluctuations on vicinal surfaces, which had been extensively examined in the continuous limit by resorting to Langevin dynamics. The diffusivity of a cluster of radius Rr was found to be L D ¼ D R
; r

(7:61)

where the exponent L , derived for different mechanisms, agrees with the findings of Van Siclen [38]; it equaled three for atoms diffusing around the cluster edges, two if diffusion across the crystal terrace was limiting, known also as the correlated evaporation– condensation mechanism, and one for evaporation–condensation defining cluster motion. Expressions for the prefactor D in the different cases were also given. To check if this approach was valid for islands of 100 or more atoms, Monte Carlo simulations were done, which proved to be in good agreement with predictions from the continuum approach. In these investigations, the clusters were circular in shape and the atomic structure of the surface was not considered.

554

Mechanism of cluster diffusion

In the derivation of large cluster movement the structure of the cluster proved to be important, so the above derivations only partly describe the situation; the experimental data will be presented at length in Chapter 9. With some of the basics established, we will now examine the empirical data and theoretical speculations about diffusion of clusters on various surfaces, which still leave uncertainties about this subject.

References [1] D. A. Reed, G. Ehrlich, In-channel clusters: Rhenium on W(211), Surf. Sci. 151 (1985) 143–165. [2] K. Kyuno, G. Ehrlich, Diffusion and dissociation of platinum clusters on Pt(111), Surf. Sci. 437 (1999) 29–37. [3] H. -W. Fink, G. Ehrlich, Pair and trio interactions between adatoms: Re on W(110), J. Chem. Phys. 81 (1984) 4657–4665. [4] O. S. Trushin, P. Salo, T. Ala-Nissila, Energetics and many-particle mechanisms of twodimensional cluster diffusion on Cu(100) surfaces, Phys. Rev. B 62 (2000) 1611–1614. [5] D. A. Reed, G. Ehrlich, One-dimensional random walks of linear clusters, J. Chem. Phys. 64 (1976) 4616–4624. [6] U. M. Titulaer, J. M. Deutch, Some aspects of cluster diffusion on surfaces, J. Chem. Phys. 77 (1982) 472–478. [7] J. D. Wrigley, D. A. Reed, G. Ehrlich, Statistics of one-dimensional cluster motion, J. Chem. Phys. 67 (1977) 781–792. [8] W. R. Graham, G. Ehrlich, Surface diffusion of atom clusters, J. Phys. F 4 (1974) L212–214. [9] T. R. Linderoth, S. Horch, L. Petersen, S. Helveg, E. Laegsgaard, I. Stensgaard, F. Besenbacher, Novel mechanism for diffusion of one-dimensional clusters; Pt/Pt(110)(1 × 2), Phys. Rev. Lett. 82 (1999) 1494–1497. [10] F. Montalenti, R. Ferrando, Leapfrog diffusion mechanism for one-dimensional chains on missing-row reconstructed surfaces, Phys. Rev. Lett. 82 (1999) 1498–1501. [11] J. D. Wrigley, G. Ehrlich, Two-dimensional random walks of diatomic clusters, J. Chem. Phys. 84 (1986) 5936–5954. [12] C. M. Chang, C. M. Wei, S. P. Chen, Modeling of Ir adatoms on Ir surfaces, Phys. Rev. B 54 (1996) 17083–17096. [13] A. Bogicevic, J. Strömquist, B. I. Lundqvist, Low-symmetry diffusion barriers in homoepitaxial growth of Al(111), Phys. Rev. Lett. 81 (1998) 637–640. [14] S. C. Wang, G. Ehrlich, Diffusion of large surface clusters: Direct observations on Ir(111), Phys. Rev. Lett. 79 (1997) 4234–4237. [15] S. C. Wang, U. Kürpick, G. Ehrlich, Surface diffusion of compact and other clusters: Irx on Ir (111), Phys. Rev. Lett. 81 (1998) 4923–4926. [16] U. Kürpick, B. Fricke, G. Ehrlich, Diffusion mechanisms of compact surface clusters: Ir7 on Ir (111), Surf. Sci. 470 (2000) L45–51. [17] Z. -P. Shi, Z. Zhang, A. K. Swan, J. F. Wendelken, Dimer shearing as a novel mechanism for cluster diffusion and dissociation on metal (100) surfaces, Phys. Rev. Lett. 76 (1996) 4927–4930. [18] G. L. Kellogg, Oscillatory behavior in the size dependence of cluster mobility on metal surfaces: Rh on Rh(100), Phys. Rev. Lett. 73 (1994) 1833–1836.

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[19] V. Chirita, E. P. Münger, J. E. Greene, J. -E. Sundgren, Reptation: A mechanism for cluster migration on (111) face-centered-cubic metal surfaces, Surf. Sci. 436 (1999) L641–647. [20] J. C. Hamilton, M. S. Daw, S. M. Foiles, Dislocation mechanism for island diffusion on fcc (111) surfaces, Phys. Rev. Lett. 74 (1995) 2760–2763. [21] J. C. Hamilton, Magic size effects for heteroepitaxial island diffusion, Phys. Rev. Lett. 77 (1996) 885–888. [22] S. Aubry, P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground states, Physica (Amsterdam) D 8 (1983) 381–422. [23] J. C. Hamilton, M. R. Sorensen, A. F. Voter, Compact surface-cluster diffusion by concerted rotation and translation, Phys. Rev. B 61 (2000) R5125–5128. [24] J. Zhuang, Q. Liu, M. Zhuang, L. Liu, L. Zhao, Y. Li, Exchange rotation mechanism for dimer diffusion on metal fcc(001) surfaces, Phys. Rev. B 68 (2003) 113401 1–4. [25] M. I. Haftel, Surface reconstruction of platinum and gold and the embedded-atom method, Phys. Rev. B 48 (1993) 2611–2622. [26] M. I. Haftel, M. Rosen, T. Franklin, M. Hettermann, Molecular dynamics observations of the interdiffusion and Stranski-Krastanov growth in the early film deposition of Au on Ag(100), Phys. Rev. Lett. 72 (1994) 1858–1861. [27] M. I. Haftel, M. Rosen, Molecular-dynamics description of early film deposition of Au on Ag (110), Phys. Rev. B 51 (1995) 4426–4434. [28] R. A. Johnson, Analytical nearest-neighbor model for fcc metals, Phys. Rev. B 37 (1988) 3924–3931. [29] Q. Liu, Z. Sun, X. Ning, Y. Li, L. Liu, J. Zhuang, Systematical study of dimer diffusion on metal fcc(001) surfaces, Surf. Sci. 554 (2004) 25–32. [30] V. Rosato, M. Guillope, B. Legrand, Thermodynamical and structural properties of f.c.c. transition metals using a simple tight-binding model, Philos. Mag. A 59 (1989) 321–336. [31] M. Guillope, B. Legrand, (110) surface stability in noble metals, Surf. Sci. 215(1989) 577–595. [32] E. Pijper, A. Fasolino, Mechanism for correlated surface diffusion of weakly bonded dimers, Phys. Rev. B 72 (2005) 165328 1–5. [33] J. -M. Wen, S. -L. Chang, J. W. Burnett, J. W. Evans, P. A. Thiel, Diffusion of large twodimensional Ag clusters on Ag(100), Phys. Rev. Lett. 73 (1994) 2591–2594. [34] A. F. Voter, Classically exact overlayer dynamics: Diffusion of rhodium clusters on Rh(100), Phys. Rev. B 34 (1986) 6819–6829. [35] J. R. Sanchez, J. W. Evans, Diffusion of small clusters on metal (100) surfaces: Exact masterequation analysis for lattice-gas models, Phys. Rev. B 59 (1999) 3224–3233. [36] P. Salo, J. Hirvonen, I. T. Koponen, O. S. Trushin, J. Heinonen, T. Ala-Nissila, Role of concerted atomic movements on the diffusion of small islands on fcc(100) metal surfaces, Phys. Rev. B 64 (2001) 161405 1–4. [37] J. M. Soler, Monte Carlo simulation of cluster diffusion in a triangular lattice, Phys. Rev. B 50 (1994) 5578–5581. [38] C. D. Van Siclen, Single jump mechanisms for large cluster diffusion on metal surfaces, Phys. Rev. Lett. 75 (1995) 1574–1577. [39] J. M. Soler, Cluster diffusion by evaporation-condensation, Phys. Rev. B 53 (1996) R10540–10543. [40] S. V. Khare, N. C. Bartelt, T. L. Einstein, Diffusion of monolayer adatom and vacancy clusters: Langevin analysis and Monte Carlo simulations of their Brownian motion, Phys. Rev. Lett. 75 (1995) 2148–2151. [41] S. V. Khare, T. L. Einstein, Brownian motion and shape fluctuations of single-layer adatom and vacancy clusters on surfaces: theory and simulations, Phys. Rev. B 54 (1996) 11752–11761.

8

Diffusivities of small clusters

In the previous chapter we presented possible mechanisms which can contribute to cluster diffusion; in this chapter, we will concentrate on the energetics of cluster movement, mostly the movement of the center of mass. Early studies of cluster diffusion were all done on tungsten surfaces using FIM, but as techniques other than field ion microscopy were applied to learning more about this subject, other surfaces came under scrutiny. The biggest change in the level of activity, however, was made by theoretical calculations. These now dominate the field and have usually covered several surfaces of different materials in one examination. The number of experimental studies of cluster behavior decreased markedly as computational efforts reached new intensities. Unfortunately, theoretical investigations still are quite uncertain and experiments are urgently needed for comparison and verification. Nevertheless we will try to arrange our comments chronologically in the description of each material, but with experiments and theoretical calculations separated.

8.1

Early investigations

8.1.1

Experiments Work on the diffusion of single adatoms on a metal surface had been going on for just a few years when Bassett began to look at clusters formed by association of several atoms [1]. In 1969 he noted that after depositing several atoms on the (211) and (321) planes of tungsten, clusters formed, with a mobility smaller than that of single atoms, provided that deposition took place with the atoms in the same channel. On these planes, clusters moved in only one dimension, along the channels of the planes. Studied were tantalum, tungsten, rhenium, iridium, and platinum atoms. On the (110) plane of tungsten behavior was different – as one can expect from the structure of this plane, movement was twodimensional. Tantalum formed stable two-dimensional clusters of several atoms, with dimers having a mobility roughly one tenth that of singles. As shown in Fig. 8.1, atoms in platinum and iridium dimers were resolved in the FIM, and were oriented along the closepacked directions of the surface. They diffused only slightly less rapidly than individual adatoms. Easily formed dimers turned out not to be typical for all metallic adatoms: rhenium adatoms did not form dimers at all, and only bigger Re clusters were observed. This study showed, however, that larger clusters could be built up one atom at a time with very precise control.

8.1 Early investigations

557

Fig. 8.1

Three dimers and one trimer (nearest center) are seen in FIM on W(110) plane after allowing Pt atoms deposited on the surface to diffuse at 240 K (after Bassett and Parsley [1]).

Fig. 8.2

Formation of Ta5 cluster on W(110) surface observed in FIM. Five Ta atoms were deposited on (110) plane at 77 K in (a). After 2 min at 291 K, pentamer has formed in (b) (after Bassett [2]).

At the beginning of 1970 Bassett [2] followed his first investigation with a number of small descriptions of various clusters. A two-dimensional tantalum pentamer on W(110), shown in Fig. 8.2, was created as well as a rhenium and a linear iridium heptamer [3]. A much more detailed description of the technique for making and observing clusters followed, showing the creation of a hexamer of iridium on W(110), in Fig. 8.3, as well as rhenium and tantalum clusters [4]. The diffusion of clusters was mentioned only briefly, however. Dimer mobility was noted as being smaller than that of adatoms by a factor ranging from two to ten, but tantalum dimers had a mobility comparable with that of tantalum adatoms.

558

Diffusivities of small clusters

Fig. 8.3

Steps in the formation of a linear Ir6 cluster on W(110) examined in FIM, starting with an initial deposition in (a) on surface cooled to 78 K. (b) After heating to 380 K for 1 min. (c) After an additional 3 min heating. (d) After a further 4 min heating a cluster has formed (after Bassett [4]).

A year after Bassett, Tsong [5,6] looked at W-Re, as well as W2 dimers on W(110) and Re3 as well as ReW2 trimers, and found that for a W-Re hetero-dimer at temperatures between 300 and 340 K the tungsten atom can migrate away, while the Re atom stays stationary. A stable Re-W cluster, as well as a Re2 cluster with atoms in nearest-neighbor positions was not observed below 332 K. However, a rhenium atom was found to easily combine with a cluster of two tungsten atoms, which is stable up to 460 K, and starts migration above 380 K. Clusters made up of three Re atoms were also stable on this plane. The situation differs for tungsten clusters. W dimers were easily created and stable up to 390 K, with dimer atoms always observed at the nearest-neighbor distance. Tungsten clusters with up to six atoms dissociated below 490 K. The energetics of cluster movement was not investigated in this study. Tsong [5,6] also examined a chain of four tungsten atoms in adjacent channels of the W(211) plane, which was observed to migrate as a unit. Somewhat later there followed a further description of clusters on tungsten [6]. A tungsten dimer with atoms in adjacent rows was observed to migrate as a unit on W(211) at up to 330 K and a tetramer with atoms in adjacent rows was seen to diffuse at 300 K. Some of these clusters are shown in Fig. 8.4. When a rhenium atom was present in a channel next to tungsten, the two migrated together but dissociated frequently even below a temperature of 320 K. When both atoms were present in the same channel they formed a stable Re-W hetero-dimer, with much lower mobility. Two Re adatoms deposited in the same channel also readily combined to form a Re2 dimer, which behaves very similarly to a dimer of tungsten. Also examined were the structures of various clusters. Two configurations were identified for W2 on W(211), and possible structures were proposed for W2, W3, W4, and W6, as well as for rhenium clusters on the W(110) plane; the latter have to be looked at as speculative. What is clear from this study is that creating clusters depends on the structure of the surface. These early studies were quite qualitative; they provided only little information about the migration of clusters, and were more concerned with cluster structure. A break in this trend was made by Bassett and Tice [7], who did quantitative studies of cluster stability, but that is not our primary concern here.

559

8.1 Early investigations

Fig. 8.4

Tungsten clusters on W(211) plane observed in FIM. (a) Ten tungsten atoms were deposited on W(211) plane. On heating to 300 K, six combined to form W6, the rest combined into linear W4 in (b). (c)–(f) Displacements of clusters on heating to 300 K (after Tsong [6]).

8.1.2

Theoretical investigations Calculational efforts began with high intensity in the early 1990s. As will quickly become apparent, these studies nicely supplement experimental work in providing detailed information not otherwise accessible. The first molecular dynamics simulations of clusters were done by Tully et al. [8] on an fcc(100) surface using a ghostparticle theory to avoid dealing with the many surface and bulk atoms. Adatoms interacted with each other via Lennard-Jones potentials, and the crystal was not allowed to relax. From an Arrhenius plot of the diffusivities in Fig. 8.5 they were able to arrive at the diffusion parameters shown in Table 8.1. From dimers upward there was a gradual increase in the activation energy, and generally a decrease in the Table 8.1 Rate parameters for diffusion of Lennard-Jones Pt clusters on fcc(100) [8]. Cluster size N

Prefactor V0×1012

Activation Energy ED (eV)

1 2 3 4 5 6

7.62 7.34 6.58 6.84 5.57 4.30

0.390 ± 0.005 0.338 ± 0.007 0.360 ± 0.010 0.415 ± 0.013 0.418 ± 0.013 0.425 ± 0.013

560

Diffusivities of small clusters

100

Diffusivity D

10–1

N=1 10–2 2 3

10–3

4 5 6

10–4

0

1

2

3 1/T

4

5

6

Fig. 8.5

Arrhenius plots for the diffusion of differently sized clusters (after Tully et al. [8]) interacting via Lennard-Jones potentials.

Fig. 8.6

Trajectory of dimer at T = 0.34 εLJ =k during multiple jumps on fcc(111) surface obtained in molecular dynamics simulations with Lennard-Jones potentials (after Ghaleb [9]).

prefactor, as the size of the cluster increased from two up to six atoms. The energy unit employed in the molecular dynamics simulations here was 0.25 eV; distances were measured in terms of the spacing of the Lennard-Jones minimum, 1.39 Å. It must be noted that for platinum the unit of temperature was 2910 K, so that diffusion parameters were obtained over a very wide energy range. In 1984, Ghaleb [9] did computer simulations of dimer diffusion on an fcc(111) crystal also relying on Lennard-Jones potentials and came up with very interesting results. At an elevated temperature of 0.34 εLJ =k, where εLJ is the depth of the Lennard-Jones potential, both atoms were found to jump simultaneously and to execute long transitions,

8.2 Clusters on aluminum surfaces

561

indicated in Fig. 8.6. This was a surprising and most important result, which changed how cluster diffusion was interpreted. At a lower temperature of 0.22 εLJ =k transitions were more usual; they occurred between nearest-neighbor adsorption sites, through jumps of one atom to an adjacent site, and the mobility of the center of mass of dimers was very close to the mobility of single atoms. With the early cluster studies out of the way, we will now begin listing work according to the surface examined.

8.2

Clusters on aluminum surfaces

8.2.1

Aluminum (100) There is no experimental information available about the energetics of cluster movement on aluminum surfaces. Theoretical studies started with local-density-functional calculations for the diffusion of a pair of aluminum atoms on Al(100), done by Feibelman [10] in 1987. For single Al adatoms he arrived at a barrier to surface diffusion of 0.80 eV. The activation energy for the dimer was lower, however, amounting to only 0.66 eV. In the dimer, the two adatoms were found to be sitting higher from the surface than does a single atom, and the bonding to the surface was somewhat diminished as the two adatoms interacted. Diffusion occurred by the jumping of one atom at a time, so the distance between the two adatoms increased, and the atom not jumping could strengthen its interactions with the lattice. The energy gained was transferred to the moving adatom, lowering the diffusion barrier. This reduction in the activation energy for dimer movement compared to that of a single adatom was also found on Ni(100) and Pt(100) [11]. Further estimates for diffusion of aluminum clusters, illustrated in Fig. 8.7, were done by Valkealahti and Manninen [12] with effective medium theory. Diffusion of dimers on the (100) face was slower than on the (111) plane, and close to 600 K exchange events occurred. Dimers were not dissociated below 600 K, and trimers behaved similarly. No energetics for dimers were derived, however. Bogicevic et al. [13] used the VASP simulation package, as well as DACAPO [14] to look at self-diffusion of Al, Au, and Rh dimers on (100) planes. In this study, contrary to the findings of Feibelman [10], dimer atoms occupy neighboring hollow sites. A number

Fig. 8.7

Aluminum dimer on aluminum clusters, calculated with EMT. (a) On Al(100). (b) On Al(111) (after Valkealahti and Manninen [12]).

562

Diffusivities of small clusters

–0.04

0.54

0.96 1.16 1.28 0.96 1.28 1.16 0.53 0.23

0.83 0.53

1.39 1.09 1.38 0.55

1.07 1.36

–0.03

–0.04

–0.03

Fig. 8.8

Activation energies (in eV) for diffusion and dissociation of Al2 on Al(100) plane obtained from density-functional calculations. Binding energies are given at top or bottom of each configuration (after Bogicevic et al. [13]).

Fig. 8.9

Diffusion of copper pentamer on Cu(100) plane. (a) Internal three atom shearing. Activation energies (in eV) from EAM calculations. (b) Dimer rotation. (c) Schematic for diffusion of Al7 cluster on Al(100) plane (after Trushin et al. [15]).

of processes was considered; two of them seemed to be more important than the others: shearing (middle of the second row in Fig. 8.8 and to the left) as well as stretching (the middle row, center to right). The process described by Bogicevic et al. [13] as shearing is in fact that of a dimer atom moving perpendicular to the dimer axis, which can be achieved by an atom jump to the side. For aluminum dimers the barrier for such described shearing (0.53 eV) was 0.3 eV lower than for stretching (0.83 eV). It appears that for these systems this process is the low-energy path for the dimers. The authors argue that this is likely to be a general effect for other (100) fcc metals. Slightly bigger clusters than dimers were investigated by Trushin et al. [15] on Al(100) with the Ercolessi–Adams [16] glue potential. The new processes considered, internal three-atom shearing, row shearing at the edges, and dimer rotation, are all illustrated in Fig. 8.9. Here by shearing the authors mean the movement of two (dimer shearing) or three atoms (trimer shearing) together in concerted motion. With Al7 clusters they found

563

8.2 Clusters on aluminum surfaces

Table 8.2 Activation energies (in eV) for dimer diffusion evaluated by density-functional calculations on fcc(100) surfaces [20].

Dimer

ED(Jump) ED(Ex) ED(expt)

Al

Rh

Ir

Ni

Pd

Pt

Cu

Ag

Au

0.54 0.40

1.33 1.61

1.87 1.30 0.88

0.88 0.69

0.86 0.75

1.30 0.18 0.41

0.66 0.77

0.55 0.69

0.82 0.24

that dimer shearing at the edges occurred over a barrier of 0.44–0.45 eV, and internal three-atom row shearing occurred over 0.42 eV, small compared to the activation energy for atom jumps of 0.53 eV and 0.56 eV for exchange. The energetics of Al7 diffusion are illustrated in Fig. 8.9; exchange events of low energy were not found. Molecular dynamics simulations of dimer self-diffusion have been carried out by Zhuang et al. [17] on the (100) surface of aluminum using the EAM potentials of Johnson [18]. The mechanism for dimer diffusion preferred was the conventional exchange process with a value of 0.342 eV for the barrier compared to 0.425 for conventional hopping. More complicated mechanisms of movement, such as 180° and 270° exchange processes were also observed for aluminum dimers, however, less frequently and their energetics were not revealed. Very similar findings were reported by Liu et al. [19]. The results for static activation energies are given in Table 7.3 and hopping has to proceed over a higher barrier (0.4614 eV) than exchange (0.3545 eV). More complicated mechanisms on this surface were also observed. In 2005 Chang and Wei [20] used a different theoretical approach and carried out ab initio density-functional calculations using the VASP [21,22] package to establish the activation energy for self-diffusion of dimers on different fcc(100) planes. Their results are given in Table 8.2, calculated with a (5 × 5) surface cell using the LDA approximation and the Ceperley–Alder potential [23,24]. Diffusion by atom exchange was predicted for aluminum with an activation energy of 0.40 eV compared to 0.54 eV for simple hopping. More complicated mechanisms were not taken into account. Unfortunately there is no experimental information for movement of clusters on aluminum and the theoretical investigations are not conclusive. The activation energies for movement of dimers by hopping are quite spread out, and range from 0.425 eV to 0.66 eV. However, most investigations indicate exchange as a leading mechanism and three values obtained so far are not too far from each other: 0.34 eV, 0.35 eV, and 0.4 eV [17,19,20]. Only Trushin [15] looked at diffusion of clusters bigger than a dimer on Al(100) and concluded that shearing played an important role.

8.2.2

Aluminum (111) Investigations of the motion of aluminum dimers on Al(111) started with the study of Bogicevic et al. [25] using density-functional theory. Both intra- and inter-cell movements were considered. In the former, the dimer is confined to an hexagonal cell containing six sites around one surface atom and long-range diffusion does not take place, whereas in the latter case movement over long distances is accessible. These steps of

564

Diffusivities of small clusters

Intra-cell motion

hh E (eV) 0.08

ff

fh

hh

fh

Inter-cell motion

bb

fh E (eV)

atom-by-atom rotation

0.04

0.25

0 concerted sliding (a)

Fig. 8.10

0.50

concerted rotation

0 atom-by-atom rotation (b)

concerted sliding

Motion of aluminum dimer on Al(111) derived from density-functional calculations. (a) Confined (non-diffusive) motion. (b) Inter-cell diffusion (after Bogicevic et al. [25]).

dimer movement are illustrated in Fig. 8.10. Dimer motion for aluminum is different from what was reported for Ir2 on Ir(111) [26], where single atom jumps were assumed. With aluminum there is a strong attraction between the two atoms, and rotation out of the cell requires overcoming a barrier of 0.50 eV. Once done, the second step occurs without any barrier, and the dimer moves by concerted sliding. Concerted inter-cell sliding in the direction required only 0.13 eV. In intra-cell rotation by individual atom jumps over the bridge, the highest barrier found was 0.03 eV. Rotational motion of the dimer already began at 8 K, but leads only to local movement. Concerted intra-cell rotation of the dimer required three times more energy, 0.09 eV, as estimated in the GGA approximation. Concerted intra-cell sliding occurred over a barrier of 0.08 eV. The effect of compressive strain, a reduction in the lattice spacing, is to significantly interfere with dimer diffusion, as the energy of dimers on a pair of adjacent fcc or hcp sites is increased. As is clear from the energetics, intra-cell movement will proceed mostly through atom by atom transitions. However, for longer-range diffusion, the dimer must overcome a bigger barrier than for local movement. Valkealahti and Manninen [12] investigated diffusion of aluminum dimers on a polyhedron, illustrated in Fig. 8.7, with effective medium theory. On the (111) plane, dimers were quite mobile starting at 100 K, and did not dissociate below 600 K. At higher temperatures exchange events occurred with cluster atoms. Self-diffusion of metal clusters with up to five atoms was examined by Chang et al. [27] with density-functional theory in the local-density approximation (VASP). For the aluminum dimer Al2 on Al(111), the lowest energy was obtained for a “mixed fcc-hcp” dimer with one atom in an hcp site and the other in an fcc position. Movement with the lowest activation energy occurred via zigzag motion illustrated by the sequence a1 ↔ a2 ↔ a3, a4, or a5, in Fig 8.11a, with only one atom moving at a time. Zigzag transitions result in local movement which can occurr by the translation a1 ↔ a2 ↔ a3, or by dimer rotation a1 ↔ a2 ↔ a4 or a5. Local motion had a small activation energy of 0.05 eV, local movement of two atoms together required 0.09 eV, making it rather unlikely. Long-range diffusion (movement out of a cell) meant overcoming a barrier of 0.08 eV. Details of Al3 diffusion over the Al(111) surface have also been studied by Chang et al. [28]. The cluster can exist in a number of different configurations on the surface, of which

8.2 Clusters on aluminum surfaces

Fig. 8.11

565

Structure and diffusion path for Al clusters on Al(111) surface. Energetics from EAM calculations for dimer on Al(111). Numbers indicate structural energies in eV (after Chang et al. [27]).

the configuration of the lowest energy is a triangular trimer in fcc sites with the center over an empty site, designated FCC-H; a final letter T indicates that the cluster center is on top of an surface atom. It should be noted that this was also the preferred arrangement in experiments with iridium trimers on Ir(111) [26]. The possible arrangements are shown in Fig. 8.12, L indicating linear, and it is clear that the linear clusters have a higher energy, as do non-compact triangular trimers, shown in Fig. 8.12c. Three different non-local diffusion steps were identified: concerted translation between FCC-H to HCP-T with an energy of 0.24 eV, translation between FCC-T to HCP-H, with energy of 0.22 eV, and transformations between compact triangular clusters and linear trimers with an activation energy of 0.21 eV. Translation of linear trimers proceeds over a higher barrier of 0.28 eV. The authors conclude that the most likely diffusion path for aluminum trimers is that shown in Fig. 8.12a, since it involves the largest displacements with the lowest number of steps. This path involves transitions of a compact trimer via a jump to an HCP site over a lattice atom, then rotation to an FCC site, and finally a jump to an HCP-H site. A variety of other migrations were considered with this as the most probable one, which takes place by concerted

566

Diffusivities of small clusters

Fig. 8.12

Al3 configurations on Al(111) surface studied with DFT. (a) Structure and diffusion path for concerted movement of triangular Al3 on Al(111). Number over arrow gives transition state energy in eV. (b) Linear trimer. Number under structure gives calculated total energy (in eV) compared to most stable. (c) Non-compact triangular trimers (after Chang et al. [28]).

jumps and rotation of the whole trimer. This was confirmed by molecular dynamics simulations based on the embedded atom method, which also detected the trimer rotation HCP-T ↔ FCC-T. The study sugests that trimers exist mostly in the compact triangular form, and it would be crucial to experimentally check the presence of linear trimers on the Al(111) surface. Diffusion of larger clusters was also considered, the most interesting being the tetramer, in Fig. 8.11b. There the lowest energy path was observed by movement between compact configurations. Translation took place by the transition b1 ↔ b2 ↔ b3, and rotation via b1 ↔ b2 ↔ b4 or b5. The concerted intra- and inter-cell motion of tetramers is more demanding than zigzag motion with two atoms moving together. The activation energy for long-range Al4 diffusion was only 0.18 eV, but increased again for pentamers, where it was 0.30 eV for movement of two pentamer atoms from fcc sites to adjacent hcp sites or 0.37 eV for motion from hcp to fcc sites. For pentamers the most favorable process was rotation with only two atoms moving (dimer shearing), c1 → c2 → c3 in Fig 8.11c. Long-range concerted motion required more energy of 0.45 eV for movement between fcc to hcp sites or 0.49 eV for the opposite direction. Self-diffusion barriers were also calculated for other metals with EAM potentials and are shown in Fig. 8.13. What is

567

8.3 Clusters on iron surfaces

0.4 0.3 0.2 0.1

Al (DFT)

Al (EAM)

Activation energy ED(eV)

0.3

0.3

0.2

0.2

0.1

Ni (EAM)

Pd (EAM)

0.6 0.4 0.2 Pt (EAM) 0.20 0.15 0.10 0.05 0.8 0.6 0.4 0.2

Ag (EAM)

Ir (EAM) 1

Fig. 8.13

0.20 0.15 0.10 0.05

2

Cu (EAM)

Au (EAM)

Ir (expt)

3 4 5 1 2 3 4 Number of atoms in cluster

0.1 0.25 0.20 0.15 0.10 0.05 0.4 0.3 0.2 0.1 0.6 0.5 0.4 0.3 0.2

5

Self-diffusion barriers calculated on fcc(111) planes of various metals with EAM potentials (after Chang et al. [27]).

important here is that all show a decrease in the diffusion barrier in going from trimers to tetramers, and then increase again for pentamers, as found a little earlier for Ir(111) by Wang [29]. Unfortunately we have no experimental information for clusters on Al(111). The theoretical predictions agree that there is no exchange involved in the movement of clusters, and a number of correlated and uncorrelated motions were studied. The low energy for rotation of dimers seems to be established. Longer movement of dimers is more demanding. Only Chang et al. [27] looked at the change in the energetics with the size of clusters and noticed a minimum for the energy of tetramer diffusion. Further experimental investigations are definitely in order to uncover the true nature of cluster movement on this plane.

8.3

Clusters on iron surfaces In 2007, Chen et al. [30] calculated the diffusivity of iron clusters on Fe(110), using modified analytical EAM potentials [31,32]. Binding of the cluster is greater for islands than chains, as is clear from Fig. 8.14. They evaluated the mean-square displacement at different temperatures, and from Arrhenius plots of the diffusivity were able to arrive at

568

Diffusivities of small clusters

Table 8.3 Diffusion characteristics for iron clusters on Fe(110) from EAM calculations [30].

Cluster size (atoms)

Activation energy ED (eV)

Prefactor for diffusivity Do (cm2/sec)

1 2 3 4 5 6 7 8 9

0.40 0.36 0.49 0.84 0.71 0.75 1.16 0.89 0.65

1.27 × 10−2 1.59 × 10−3 2.81 × 10−3 1.89 × 10−1 7.71 × 10−3 2.98 × 10−2 1.01 × 10−1 1.18 × 10−2 6.60 × 10−4

0 Fe island Fe chain

–0.32

Binding energy (eV)

–1

–0.84 –1.41

–2

–1.90 –2.47

–3 –3.20 –3.76

–4

–4.32

–5 2

Fig. 8.14

4

6 Number of atoms

8

10

Energy of iron chain compared with that of islands obtained in EAM calculations on Fe(110) (after Chen et al. [30]).

the activation energies and prefactors, listed in Table 8.3 for clusters made up of up to nine atoms. It is of interest that the tetramer and heptamer have unusually high activation energies as well as prefactors. Movement of dimers required a lower energy than monomers. Chen et al. concluded that dimer movement proceeds by sequential jumping of dimer atoms and that the lower energy is due to the higher coordinational number. For bigger clusters, the sequentional displacement of atoms along cluster edges was suggested and the increasing mobility of non-compact clusters was attributed to diffusion of the extra atom along the periphery. Regrettably there are no measurements available to validate these estimates.

569

8.4 Clusters on nickel surfaces

8.4

Clusters on nickel surfaces

8.4.1

Nickel (100) For nickel just as for aluminum surfaces we have no experimental information about cluster movement. Investigations on Ni(100) started with calculations using embedded atom potentials by Liu and Adams [11], done for clusters both on the (100) and (111) planes of nickel. The EAM potentials have not always given good agreement with experiments but show two-dimensional islands for all cluster sizes on Ni(100), and dissociation energies oscillating with size. However, as can be seen in Table 8.4, on the (100) plane Liu and Adams observed an oscillation in the diffusion barrier with size different from that for self-diffusion on the (111) plane. Surprisingly, the migration energy for Ni and Pt dimers is slightly lower than for single atom movement. Table 8.4 Cluster diffusion parameters calculated with EAM potentials [11]. Activation energy ED(eV) Cluster

Ni/Ni(100)

Pt/Ni(100)

Pt/Pt(100)

Ni/Ni(111)

Ir/Ir(111) [26]

Single Dimer Trimer Tetramer Pentamer

0.63 0.61 0.68 1.00 0.68

0.76 0.69 0.81 1.00 0.78

0.47(0.64) 0.41(0.52) 0.49(0.65)

0.056 0.12 0.20 0.21 0.26

0.27 0.43 0.63 0.46 0.66

Energetics in parenthesis from Kellog [33].

The authors looked at the details of Ni and Pt dimer movement on the Ni(100) surface, the energetics of which are illustrated in Fig. 8.15a2–c2. In the initial movement of the dimer there are two options for the atomic jump. Energetically less demanding is the jump in the direction perpendicular to the dimer axis, which costs 0.61 eV for Ni and 0.69 eV for a Pt atom. A more demanding jump is in the direction of the dimer axis, which required 1.01 eV for Ni and 1.12 eV for Pt. Subsequent jumps of the second dimer atom occurs with a much lower energy of 0.25 eV for Ni and 0.43 eV for Pt. For trimers the lowest energy transition is rotation, 0.26 eV for nickel and 0.38 eV for platinum, which involves changes between the triangular forms. Since such transitions do not shift the center of mass significantly, long-range movement proceeds through the linear trimers. The optimal path for such a transition is presented in Fig. 8.15a3–d3. The most demanding step of this path is the initial one of breaking the triangular trimer, costing 0.68 eV for nickel and 0.81 eV for platinum. The initial step of breaking a compact shape is also most demanding in tetramer movement, 1.00 eV for both nickel and platinum, shown in Fig. 8.15a4–d4. For the pentamer, the energy of moving atoms is lower than for the tetramer and comparable with trimer motion, as is clear from the optimal path illustrated in Fig 8.15a5–d5. The investigations did not consider dimer shearing as a possible mechanism, which might play quite an important role, especially

570

Diffusivities of small clusters

Fig. 8.15

Schematic of optimal diffusion paths and energetics for Ni and Pt clusters on Ni(100) plane, based on the work of Liu and Adams [11] using EAM potentials.

in the movement of tetramers or pentamers. What is surprising is that for Pt clusters on Ni(100), Liu and Adams did not observe exchange, since Kellogg [34] showed that for single atom movement of Pt on Ni(100) exchange was the basic mechanism of transport. A new mechanism for movement on Ni(100) was discovered by Shi et al. [35] using EAM interactions and bond counting. They found that displacements by concerted motion of two or three atoms can be more favorable than movement by single atoms. The mechanism is illustrated in Fig. 7.25, and the energetics of the different steps in the movement of a Ni cluster are summarized in Table 7.1. For nickel tetramers the initial single jump required 0.989 eV, while the initial step by dimer shearing needed only 0.842 eV. For a nine-atom cluster, trimer shearing seems to be more favorable than movement by single atoms, at 1.051 eV compared to 1.215 eV.

8.4 Clusters on nickel surfaces

571

Further theoretical investigations of dimers on Ni(100) were done by Liu et al. [19] who relied on EAM interactions [18] in their simulations. They found that movement of nickel dimers proceeded by hopping rather than by exchange. The energy for hopping of the dimer was 0.561 eV, and for exchange 1.349 eV. A year later Chang and Wei [20] used the VASP [21,22] simulation package to also find the diffusion barrier and mechanism of single atoms and dimers. Their results for dimers are presented in Table 8.2 and for nickel dimers they suggest exchange as a leading mechanism, with an activation energy 0.69 eV, rather than hopping over a higher barrier of 0.88 eV. This investigation seems to contradict the previous findings of Liu et al. [19]. Recently Kim et al. [36] used action-derived molecular dynamics, with interactions described by a tight-binding second-moment approximation, to look at cluster movement for a range of materials. For Ni dimers on a Ni(100) surface, they found hopping in the direction of the dimer axis needed 0.747 eV, hopping perpendicular to the dimer axis required 0.376 eV; exchange of the dimer with the substrate was unlikely and required 1.351 eV. What is very surprising is that the jump of an atom in the direction perpendicular to the dimer axis has exactly the same energy as calculated by them for the jump of a single isolated adatom, making this investigation a bit suspicious. They also looked at the energetics of atom movement for bigger clusters. In their study they only considered a movement of trimer atoms from the linear state along the [110] direction. Jumps in the direction [110], the direction of the trimer axis, were contemplated and required 1.109 eV. It is clear that it is not the direction of lowest energy. In contrast, exchange in the [100] direction required 1.371 eV. A similar situation holds for clusters of four atoms. They again started only as linear clusters, even though the close-packed tetragon is usually observed on fcc(100) planes. They looked only on the energetics of jumps in the [110] direction, which is the direction of the linear tetramer. Such jumps require 0.748 eV. Exchange in the [001] direction was found to need 1.347 eV. The investigations of clusters on the Ni(100) surface are far from conclusive. Two investigations suggest hopping as the primary mechanism, while one favors exchange. Here it is worth noting that for single atoms the movement by atom exchange was suggested in experiments by Fu and Tsong [37]. The activation energies for hopping of dimers are spread out from 0.56eV to 0.88 eV, while for exchange they range from 0.69 eV to 1.35 eV. Bigger clusters were only investigated by Liu et al. [19], Shi et al. [35], and Kim et al. [36]. Shi et al. showed that dimer and trimer shearing might play a very important role in such movement; studies of Liu et al. and Kim et al., even though done later, did not take shearing into account.

8.4.2

Nickel (111) Theoretical work on the Ni (111) surface started together with the study on Ni(100) by Liu and Adams [11] using EAM potentials. On Ni(111), just as on Ni(100), the dissociation energies oscillated with cluster size. Oscillations were also observed in the diffusion energies, as is apparent from Table 8.4. The activation energy for movement of dimers increased to 0.12 eV, and a further rise to 0.20 eV was observed for trimers. For tetramers the energy was almost the same, 0.21 eV, but increased further to 0.26 eV for pentamers.

572

Diffusivities of small clusters

Fig. 8.16

Schematics of optimal diffusion path of nickel clusters on Ni(111) plane, together with EAM energetics of atom jumps, based on studies by Liu and Adams [11].

Fig. 8.17

A

B

C

D

Mechanisms of diffusion for Ni4 cluster on Ni(111) surface, based on Hamilton et al. [38] using EAM potentials. A shows Ni atoms moving almost simultaneously through bridge sites. B and D illustrate dislocation process. In B two atoms move to produce the dislocation in either C or D; this can relax following arrows.

Optimal migration paths for clusters are shown in Fig. 8.16. For dimers the initial atom jump out of the cluster required 0.032 eV, while the following transition of the second atom needed only 0.029 eV. Collective jumping from fcc to hcp sites and vice versa required 0.12 eV and 0.115 eV, making collective jumps unlikely. The authors found that the collective steps between fcc and hcp sites are the most likely mechanism of movement for bigger clusters. For collective movement of trimers an energy of 0.20 eV from fcc, or 0.187 eV from hcp, is required. The values in Fig. 8.16d–e show the energetics of possible initial jumps in atom-by-atom movement for tetramers and pentamers, the energy for the collective motion of the tetramer from fcc to hcp sites is in a comparable range and amounts to 0.21 eV, while for the pentamer 0.26 eV was required. Hamilton et al. [38] in 1995 came up with a new mechanism for cluster diffusion on fcc(111) surfaces – the nucleation of misfit dislocations. Working with nickel islands on Ni(111) surfaces and EAM potentials, they did molecular dynamics at temperatures from 400 K to 1200 K. For a tetramer they came up with the configurations shown in Fig. 8.17. All the atoms can move, as in A, to give a gliding transition. However, an alternative is also possible, and is suggested by B, where two atoms translate, and create a dislocated

8.4 Clusters on nickel surfaces

573

structure that can then continue movement in four different ways. For the gliding of a tetramer they found an energy of 0.30 eV, while for dislocation movement it was 0.36 eV, making gliding of the cluster as a whole slightly more likely. Simulations were also done for larger clusters with up to 43 atoms, and gliding as well as dislocation transitions were observed. In Fig. 7.27 is shown a dislocated cluster of 19 atoms, in which the rows of atoms can move one at a time. Hamilton et al. suggested that gliding and dislocation movement are favored over atom motion along the edges. It must be noted, however, that this mechanism does not account for the long jumps that have been observed in experiments with Ir19 clusters diffusing on Ir(111) [39]. Chang et al. [27] using EAM potentials investigated clusters up to pentamers, as shown in Fig. 8.13, deriving the activation energies for movement of the center of mass. For dimers on Ni(111), the activation energy was 0.15 eV and for trimers 0.26 eV. Movement of tetramers required a slightly lower energy of 0.24 eV than for trimers. The energy again increased to 0.29 eV for pentamers. Detailed information about the transition state for movement was provided only for aluminum clusters. Calculations of seven-atom hexagonal clusters of nickel, palladium, platinum, copper, silver, and gold on the (111) plane of the same metal were done by Longo et al. [40] using EAM potentials parameterized according to Voter and Chen [41]. They evaluated the activation energies for a cluster gliding over its own crystal, and for nickel they came up with a value of 0.58 eV. For Ni7, Pt7, and Cu7 translational motion was only observed on heating the cluster to the melting temperature. For Ni7 this was expected, as the barrier for peripheral movement was anticipated to be higher than for gliding. Recently Kim et al. [36] published their findings for clusters on Ni(111). In their investigations it is not clearly stated where the cluster is sitting in the initial stage – in fcc or hcp sites; from the choice of their hopping direction for dimers it appears that clusters are actually sitting in hcp sites. The dimer is oriented along the ½110 direction and the jump occurs along ½121, which is a jump 30° to the axis of the dimer and requires 0.128 eV. The jump at 90° to the dimer axis requires less energy, 0.087 eV. Exchange is unlikely and has a huge barrier of 1.635 eV. For trimers they considered only a triangular form. It was claimed that hopping required 0.172 eV, while exchange was unlikely at 1.873 eV. The information available for cluster movement on the Ni(111) surface is very uncertain. The findings of Liu and Adams [11] are quite similar to findings of Chang et al [27], but there is no experimental data for comparison. However, it appears that gliding as well as dislocation movement play a role in the diffusion of bigger clusters on this plane.

8.4.3

Nickel (110) One-dimensional diffusion of linear clusters of three to five atoms along the channels of fcc(110)-(2 × 1) surfaces was examined on Pt, Ag, Cu, and Ni surfaces with EAM potentials by Kürpick [42]. Four different diffusion mechanisms were considered and are indicated in Fig. 8.18. In row A, the cluster atoms are all in minimum energy positions, in row B they are at bridge sites, the transition state for concerted cluster

574

Diffusivities of small clusters

Table 8.5 Self-diffusion barriers (in eV) for clusters on (110)-(2 × 1) plane calculated with EAM for concerted motion CM, hopping onto the facet HOPup, HOPdown, and exchange Ex [42]. Values in parentheses are for unreconstructed surfaces.

Pt3 Pt4 Pt5 Ag3 Ag4 Ag5 Cu3 Cu4 Cu5 Ni3 Ni4 Ni5

A

5 4 3 2 1

CM

HOPup

HOPdown

Ex

1.55(1.48) 2.02(1.93) 2.48(2.39) 0.87(0.72) 1.15(0.96) 1.43(1.19) 1.03(0.82) 1.37(1.09) 1.71(1.37) 1.44(1.16) 1.93(1.54) 2.41(1.92)

0.98 0.98 0.98 0.62 0.62 0.62 0.78 0.77 0.77 1.06 1.05 1.04

0.70 0.67 0.67 0.38 0.37 0.37 0.46 0.43 0.43 0.65 0.62 0.60

1.30 1.36 1.36 0.74 0.75 0.76 0.87 0.89 0.89 1.19 1.21 1.22

B

5 4 3 2 1

C

D

5 4 3 2 1

5 4 3 2 1

y

x Fig. 8.18

A–D indicate different positions calculated for pentamer with EAM in diffusion on fcc(110)-(2 × 1) plane, starting in minimum energy. B gives transition state for concerted atom motion, C shows minimum energy path for leapfrog transition. D indicates transition state for exchange between atom 2 and 1 (after Kürpick [42]).

motion. In row C, atom 1 climbed out of its equilibrium position and over the other cluster atoms to carry out a leapfrog transition, indicated by the black dots at the left. Finally, in row D is shown the transition state for atom exchange. Atom 2 initiates the process, moving into the next layer, and atom 1 then hops to the initial position of atom 2, which continues movement by the leapfrog mechanism and descends after atom 5. The activation energies on the different (110)-(2 × 1) surfaces are given in Table 8.5 for the various mechanisms, the figures listed in parenthesis being the barrier to concerted jumps on the unreconstructed surface. It is clear that the leapfrog mechanism requires the smallest energy, making it the leading candidate. Jump-up in the leapfrog movement is the limiting step for all the systems treated here; both exchange as well as concerted motion require a much higher energy.

8.5 Clusters on copper surfaces

575

Recently the movement of in-channel clusters on the unreconstructed Ni(110) plane was investigated by Kim et al. [36] who resorted to action-derived molecular dynamics. All the clusters they investigated had an in-channel linear arrangement. In-channel movement of Ni dimers required 0.572 eV, exchange with a wall atom was more energetically demanding at 0.730 eV, and jumps to the next channel occurred over a huge barrier of 1.954 eV. What is more surprising, for an in-channel trimer, exchange has an energy comparable to in-channel hopping, 0.609 eV compared to 0.619 eV. A similar situation was observed for an in-channel Ni tetramer; the energy for exchange amounted to 0.626 eV, while for hopping it was 0.630 eV. Jumping of clusters over a channel is unlikely for all investigated systems.

8.5

Clusters on copper surfaces

8.5.1

Experiments In 2000, Schlösser et al. [43] carried out both extensive STM measurements and calculations to better understand the diffusion of islands on Cu(111) as well as Ag(111). They investigated diffusivity as a function of island radius and discovered that it was dependent on the size of the island. For vacancy islands with a radius larger than 15 Å on the Cu(111) surface they measured an activation energy for diffusion of 0.49 ± 0.01 eV. Schlösser et al. investigated a number of different basic processes using the EMT potential of Stoltze to account for their findings and ended their work with the conclusion that “a simple interpretation of island diffusion in terms of a single rate-limiting process is not possible.” Repp et al. [44] carried out low-temperature STM studies of Cu dimers on Cu(111), addressing the question which sites were preferentially occupied. They found that there was a small energy difference of (1.3 ± 0.5) × 10−3 eV that stabilized occupation of fcc-fcc sites compared to fcc-hcp. The hcp-hcp configuration was not observed. In the temperatures accessible to their instrument below 21 K, they detected only local dimer movement, not long-range translation. They were able to trace single dimer jumps at 5 K. As indicated in Fig. 3.25, they measured the jump rates at different temperatures, and obtained a curved Arrhenius plot. At temperatures T 5 K they found that thermally assisted tunneling played a role; at higher temperatures, transitions occurred mostly by thermal diffusion over a barrier of 0.018 ± 0.003 eV with a frequency prefactor of 8 × 1011 ± 0.5 sec−1 for movement of the dimer from the fcc-fcc to the fcc-hcp configuration. Density-functional theory calculations gave a higher value of 0.028 eV for local diffusion, and also a much higher energy difference of 0.021 eV between an fcc-hcp dimer and one in the lowest energy configuration. It should be noted that the dimer as well as trimer preferred fcc sites.

8.5.2

Theoretical investigations: Cu(100) Theoretical studies started in 1994 with the investigation of Liu [45], who made estimates of the diffusion behavior of copper clusters on Cu(100) relying on EAM interactions.

576

Diffusivities of small clusters

0.15 eV 0.51 eV

(a2)

0.71 eV

(b2)

(c2)

0.19 eV 0.21 eV

0.72 eV 0.58 eV

0.51 eV 0.56 eV

(a3)

0.21 eV (b3)

(d3)

(c3)

0.71 eV

0.47 eV 0.17 eV

(a4)

(b4)

0.19 eV (c4)

(d4)

0.41 eV (c5)

(d5)

0.71 eV 0.56 eV

0.23 eV 0.41 eV 0.72 eV

(a5) Fig. 8.19

0.23 eV (b5)

0.17 eV

Low-energy configurations of small copper clusters on Cu(100) and optimal diffusion path as evaluated with embedded atom method by Liu [45].

The most stable cluster structures are indicated in Fig. 8.19a, and foreshadow how diffusion occurs. The optimal diffusion paths for dimers to pentamers are shown in Fig. 8.19a–d. Shearing was not considered, nor was the possibility of exchange. For dimers the initial step cost 0.51 eV, while the subsequent step required only 0.15 eV. For trimers the low-energy transformation is rotation of the triangular trimer at 0.19 eV, but this will not lead to long-range diffusion. The optimal long path is shown in Fig. 8.19a 3–d3 and involves transitions between triangular and linear shapes of the trimer. For tetramer movement the limiting step is breaking the tight initial shape, which required 0.71 eV; the following steps are less energetically demanding. Pentamer movement involved lower energies than for tetramers. Activation energies for cluster movement were calculated and are given

577

8.5 Clusters on copper surfaces

Table 8.6 EAM diffusion barriers for cluster migration on Cu(100) [45]. Diffusion barrier (eV)

Monomer Dimer Trimer Tetramer Pentamer

0.45 0.51 0.56 0.71 0.56

Activation Energy ED(eV)

Cluster size

0.9 0.8 0.7 0.6 0.5 0.4 2

Fig. 8.20

4 6 8 Cluster size (atoms)

10

Activation energy for diffusion of Cu islands over Cu(100) calculated using embedded atom method. Solid curve: dimer shearing. Dashed curve: sequential movement of atoms (after Shi et al. [35]).

in Table 8.6. What is striking here is that there is an oscillation of the barrier with size from trimer to pentamer, as was seen in the work of Kellogg [46]. This is easy to understand. For a cluster with an odd number of atoms there is a central core to which an extra atom is attached, but only weakly, so it can easily diffuse around the center. This, however, does not allow long-range diffusion, which would require higher energy detachment of an atom from the central part of the cluster, unless shearing of the cluster can occur. The clusters up to pentamers are quite mobile, even at room temperature. In the diffusion of clusters on surfaces most of the emphasis has been on the jumping of single atoms. However, Shi et al. [35] have suggested that on (100) surfaces shearing of two atoms, illustrated in Fig. 7.25, could make a contribution as the easiest path for clusters with an even number of atoms. Here it should be noted that similar structures were actually observed for iridium tetramers on Ir(111) [26], which diffuse unusually rapidly. The energetics of different steps in cluster movement are summarized in Table 7.1. For example, for tetramers, it was shown that the initial step by movement of a single atom required 0.815 eV, while by dimer shearing it took only 0.688 eV. For nine atom clusters, the initial step, a single atom jump, required 0.975 eV against 0.857 eV by trimer shearing. Shearing also limited the number of intermediate steps involved in diffusion. For clusters of five, seven, and ten atoms, with a peripheral atom outside the core, the initial step can proceed in the traditional way, by single atom movement, but the next steps may involve shearing. Even in this case, movement by shearing only is possible. Quantitative estimates using bond counting and embedded atom calculations indicated an activation energy for diffusion of copper clusters on Cu(100) that increases with size, as shown in Fig. 8.20. The oscillation seen there agrees nicely with the results for rhodium clusters on Rh(100) found by Kellogg [46].

578

Diffusivities of small clusters

Various estimates for the diffusion of copper dimers have been made by Boisvert and Lewis [47]. Two possibilities, atom jumps and atom exchange with the lattice were considered, both with EAM potentials and by ab initio calculations. For dimer jumps they found an activation energy of 0.48 ± 0.03 eV and a frequency prefactor of 13(× e±0.5) × 1012 sec−1; for diffusion by atom exchange the barrier was higher, 0.73 ± 0.05 eV, as well as the frequency prefactor, which amounted to 320(× e±0.8) × 1012 sec−1, both obtained from molecular dynamics simulations in the EAM approximation. With ab initio methods the activation energy turned out to be 0.57 ± 0.06 eV for jumps and for atom exchange 0.79 ± 0.15 eV. At low temperatures dimer diffusion should therefore proceed by atoms jumping. It should be noted here that the barriers calculated for jumps of single atoms, 0.49 eV by EAM and 0.52 eV by ab initio techniques, are somewhat higher than the experimental results, but comparable with the energy calculated for dimer movement. Based on their data, the authors concluded that on a Cu(100) surface, dimers contribute to mass transport as much as do single copper atoms. Monte Carlo simulations with EAM potentials were reported by Biham et al. [48] for copper clusters. The number of jumps per second at 250 K was as follows: monomers 172 ± 5, dimers 237 ± 8, trimers 5.6 ± 0.2, pentamers 1.12 ± 0.04, and heptamers 0.28 ± 0.02. Only single atom jumps, and no correlated movements were considered. Islands with 4, 6, and 8 atoms did not move at a temperature of 250 K. What is interesting here is the high rate of dimer jumps compared with single atoms and also trimers. Trushin et al. [49] have more closely examined the concerted dimer shearing mechanism for cluster diffusion proposed by Shi et al. [35]. The energetics for a number of cluster sizes, for the initial single atom movement and for dimer or trimer shearing, are shown in Fig. 8.21. For dimers they found that movement by individual atom jumps is easier than by the whole dimer gliding, requiring 0.484 eV compared to 0.820 eV. For trimers single atom jumps again seem to be the more likely initial step N=1

N=2

N=3

0.488

0.484 0.820

N=6

0.701

Fig. 8.21

0.621

0.686

N=7

0.714

0.834

N=9

0.810

0.548

N=4

0.454

N=5

0.818

0.528

0.547

N=8

0.499

0.836

0.634

0.757

0.838

N = 10

0.821

0.947

0.667

0.682

0.846

Equilibrium shape of Cu clusters on Cu(100), together with activation energy for center of mass diffusion (bold) and dissociation calculated with EAM (after Trushin et al. [49]).

8.5 Clusters on copper surfaces

Fig. 8.22

579

Center-of-mass motion calculated using EAM for hexamer on fcc(100). (a) and (b) Translation. (c) and (d) Rotation (after Trushin et al. [49]).

than dimer shearing jumps. The situation changes for tetramers, where dimer shearing provided an easier path. For pentamers both initial types of steps are equally probable. Six atoms clusters again seem to prefer dimer shearing over atom-by-atom movement and for bigger clusters trimer shearing is a likely mechanism. The authors also demonstrated how it is possible to translate and rotate a cluster via dimer shearing, as shown for a hexamer in Fig. 8.22. The possibility of internal dimer rotation is illustrated in Fig. 7.6. Salo et al. [50] now used the formalism derived by Sanchez and Evans [51] to evaluate diffusivities for copper islands on Cu(100). The diffusivity for pentamers was given by Eq. (7.58), and the energetics of different initial steps were presented in Fig. 7.37; Salo et al. also derived the equation for the diffusivity of clusters made up of five to nine atoms. The dependence of diffusivity on the size of the cluster, taking into account the shearing mechanism, was shown in Fig. 7.39. Given in this figure are the Monte Carlo simulations for single particle events based on EMT energetics, as well as the results for single atom events from the analytical theory (again relying on EMT) and also for manyatom processes (from EAM potentials). What is clear here are the oscillations in the diffusivity with the size of the cluster for all the estimates, and that the new many-atom events have not disturbed this oscillation. However, the new events considered here quantitatively increase the diffusivities. Trushin et al. [15] by taking advantage of advances in identifying saddle points for clusters have been able to establish additional opportunities for diffusion. As an example, for Cu10 in Fig. 8.23 are shown different paths, including new internal-row shearing, which have reasonably low activation energies. These constitute further opportunities for diffusion, which can take place prior to cluster dissociation. In addition, Trushin et al. also observed an interesting event for a copper adatom on top of a cluster of 17 Cu atoms on Cu(100). By internal dimer shearing near a kink site, illustrated in Fig. 8.24, the

580

Diffusivities of small clusters

Fig. 8.23

Diffusion paths, with trimer shearing, calculated with EAM for center of mass diffusion of Cu10 on Cu(100) (after Trushin et al. [15]).

Fig. 8.24

Interlayer diffusion of copper atom inside Cu17 cluster on Cu(100), accomplished by concerted dimer shearing next to kink site (after Trushin et al. [15]).

adatom incorporates into the cluster over an activation energy of 0.43 eV, which is lower than the barrier of 0.49 eV for atom jumps on the Cu(100) surface. Liu et al. [19] looked at the movement of dimers on fcc(100) with RGL interactions [52,53]. Their results for static activation energies are given in Table 7.3 for dimer hopping as well as simple exchange. For dimers on the copper(100) surface, they found that simple hopping with an activation energy 0.4307 eV was more likely than exchange, which proceeded with an energy of 0.7853 eV. They also did molecular dynamics simulations and in a total time of 2 nsec at 650 K observed 38 hops but only one exchange and no exchange rotations. Additionally they observed cooperative hopping (Fig. 7.32), where the jump of one dimer atom triggers the jump of a second atom, and rotation hopping, which is simple hopping leading to rotation of the dimer. A year later, Chang and Wei [20] carried out ab initio density-functional calculations using the VASP [21,22] package also for dimers. Activation energies calculated with a (5 × 5)

581

2 Atom 3 Atom 4 Atom 5 Atom 6 Atom 7 Atom 8 Atom

(a)

60

(b)

40 20 0 2

0

4

6

8

Size of Island in Atoms

0 –10 1.0

2.0

3.0 4.0 5.0 1000/T (K–1)

6.0

Activation Energy ED(eV)

In [rate]

20 10

Prefactor Γ0(THz)

8.5 Clusters on copper surfaces

(c)

0.7

0.6 Basham et al. Trushin et al.

0.5 0

2

4

6

8

Size of Island in Atoms Fig. 8.25

Kinetics of Cu clusters diffusing on Cu(100) plane as a function of size. (a) Arrhenius dependence for movement of center of mass of clusters. (b) Prefactor. (c) Activation energy, obtained from EAM calculations (after Basham et al. [54]).

surface cell are given in Table 8.2. No information about the mechanism was derived but for Cu dimers, diffusion by hopping was found to be more likely than by exchange, over a barrier of 0.66 eV compared with 0.77 eV. Basham et al. [54] have done self-learning kinetic Monte Carlo simulations based on optimized EAM potentials to examine diffusion of copper islands and showed that movement of the center of mass of small islands followed an Arrhenius dependence, as in Fig. 8.25a. The results for activation energies and prefactors are shown in Fig. 8.25b,c. Several things are of interest. The activation energies for diffusion in this study are in quite good agreement with the earlier work of Trushin et al. [49,50] as well as with the findings of Shi et al. [35], which took into account dimer and trimer shearing. What is noteworthy again is the oscillation of the activation energy with size, with tetramers and hexamers having large barriers, as found some time ago in experiments with rhodium clusters on Rh(100) by Kellogg [46]. Recently, an investigation of copper clusters using action-driven molecular dynamics was done by Kim et al. [36]. For dimers they looked at the energetics of the jump of a dimer in the [110] direction and found an activation energy of 0.871 eV. Perpendicular jumps are more likely at 0.477eV. Exchange required more energy at 0.686 eV. They also examined jumps of linear trimers, elongated along the [110] direction, but only considered jumps in the direction of the trimer axis, which required 1.256 eV. Exchange in the [100] direction has a lower energy of 0.712 eV, but to compare which mechanism is favorable we should know the energetics of jumps perpendicular to the trimer axis, which

582

Diffusivities of small clusters

is not given by the authors. For linear tetramers aligned in the [110] direction, the situation is similar to that described for trimers. Jumps in the [110] direction cost 1.615 eV, exchange in the [100] direction 0.718 eV. Again values are not suitable for comparison or conclusions about the mechanism of movement. There are quite a number of theoretical investigations for the movement of Cu clusters on Cu(100), with no experimental data for comparison. Most of the calculations concentrate on the energetics of certain steps during diffusion rather than on the movement of the center of mass. From all of the work it is clear that exchange is rather unlikely on this plane and that shearing definitely plays a role in diffusion. The dimer is estimated to move over the surface by simple jumps with an activation energy lying between 0.43 and 0.66 eV, not a bad agreement. The energy needed for long-range movement of trimers is in the range of 0.5–0.6 eV; for tetramers the energy for center-of-mass movement is 0.68–0.7eV, and for pentamers 0.53–0.56 eV; the energy required for diffusion of Cu6 ~0.7 eV, for Cu7 ~0.45–0.55eV, for Cu8 ~0.63–0.76 eV, for Cu9 ~0.81–0.85 eV, and for Cu10 ~0.55–0.67 eV. Almost all these data were derived in calculations based on the EAM potential, and that is worrisome. Only for dimers were ab initio methods as well as RGL potentials employed, and the results from these methods differ slightly from the rest. For dimer diffusion an energy of 0.43 eV was derived using RGL potentials, compared with 0.66 eV using VASP. Since there are no experimental values for comparison it is difficult to judge if EAM well describes diffusion on Cu(100).

8.5.3

Theoretical investigations: Cu(111) Investigations of the movement of clusters on the Cu(111) surface started with the work of Chang et al. [27] who used EAM potentials. Self-diffusion barriers are shown in Fig. 8.13. The movement of copper dimers required 0.13 eV, trimers 0.18 eV, tetramers 0.17 eV, and pentamers 0.24 eV. The detailed path of movement was investigated only for aluminum clusters. What is worth noting is essentially the same energy for movement of trimers and tetramers. The same year, Longo et al. [40] looked at the movement of a seven atom Cu cluster on the Cu(111) surface, using EAM potentials parameterized according to Voter and Chen [41]. The Cu7 cluster glided with an energy of 0.36 eV; small deformations of the cluster by translational motion from fcc to hcp sites and vice versa were observed on heating the cluster to the melting temperature; the process started at 430 K. Marinica et al. [55] in 2004 carried out extensive calculations of energy barriers for cluster diffusion on Cu(111) using embedded atom potentials devised by Mishin et al. [56]. They addressed the various configurations shown in Fig. 8.26. It should be noted that the authors have designated fcc sites by f and hcp ones by h. Their results for dimers are shown in Fig. 8.27a for intra-cell movement and in Fig. 8.27b for inter-cell diffusion. In intra-cell movement the smallest barrier of 0.016 eV occurs in atom-by-atom rotation from ff to fh, in excellent agreement with the experimental findings of Repp et al. [44] for the same kind of movement. A much higher barrier of 0.026 eV intervenes for atom-byatom rotation between fh and hh, which implies that movement backward and forward between ff and fh places is more likely than a full rotation. In inter-cell movement, by far

583

8.5 Clusters on copper surfaces

Fig. 8.26

Cluster configurations found in EAM calculations on fcc(111) plane. Black dots indicate hcp sites (after Marinica et al. [55]).

8 8 -

-

-

Fig. 8.27

-

Motion of Cu clusters on Cu(111) with accompanying energy changes. (a) Intra-cell movement of dimer. (b) Inter-cell movement of dimer. (c) Concerted motion of Cu trimer. (d) Concerted diffusion of Cu tetramer (after Marinica et al. [55]).

the lowest barrier exists for concerted sliding between ff and hh, and amounts to 0.120 eV; the same path covered in the opposite direction required only 0.039 eV. Longer atom-by-atom motion requires overcoming a much higher barrier of 0.40 eV. Trimer energetics are shown in Fig. 8.27c and have a barrier of 0.155 eV for concerted translation compared with 0.111 eV for concerted rotation. It should be noted here that

584

Diffusivities of small clusters

Table 8.7 EAM diffusion kinetics for concerted motion of clusters on Cu(111) [55]. Adisland

Cu

Barrier (eV) f→h 0.041 h→f 0.036

Cu2

Cu3

Cu4(D)

Cu4(O)

Cu7

0.132 0.120

0.155 0.133

0.262 0.238

0.189 0.165

0.388 0.348

4.51 6.47

4.12 5.05

4.65 5.70

13.17 18.00

attempt frequency (1012 sec−1) f→h 1.14 1.32 h→f 1.16 1.40

D – diagonal motion, O – oblique motion

0 monomer trimer

500 K

In Γ

–5

dimer

225 K tetramer

–10 heptamer –15

–20 0.001

0.002

0.003

0.004

0.005

1/T (K–1) Fig. 8.28

Arrhenius plots derived from EAM calculations for Cu clusters hopping from fcc to hcp sites on Cu(111) plane. Tetramer curve is for oblique transition (after Marinica et al. [55]).

atom-by-atom jumps have been disregarded as requiring too much energy; linear trimers were not considered. Energetics for tetramer motion are given in Fig. 8.27d, and reveal that what is called oblique motion as energetically preferred, with an energy of 0.189 eV from f to h and 0.165 eV from h to f. Diagonal motion, along the long diagonal of the diamond, required 0.262 eV from f to h, and 0.238 eV for the reverse process. For hexagonal heptamers the maximum barrier was found to be 0.388 eV, for movement from f to h and 0.346 eV in the opposite direction. An Arrhenius plot for the jump rates of all these clusters is given in Fig. 8.28, and shows how differently the heptamer behaves. The values of diffusion barriers and attempt frequencies for concerted motion of clusters are summarized in Table 8.7, from which the increase in attempt frequency for three atom clusters and higher is clear. Morgenstern et al. [57] did molecular dynamics simulations of the movement of Cu dimers on Cu(111) and Ag(111), shown in Fig. 8.29. The energy for self-diffusion on the

585

8.5 Clusters on copper surfaces

(a) AI: 30 Ir: 170

92 (41)

14 (23)

(c)

95 (90)

AI: 70

78 (18) (b)

92 (41)

15 (25)

AI: 30 (e)

(d) AI: 70 Ir: 300

92 (41)

(g)

AI: 130 Ir: 240

143 (121)

(f) 122 (116)

84 (112)

x(

a 10

0 (5)

AI: 220 290 340

245 260 340 300 K, and trimers above 400 K. Diffusion of Co dimers on Pt(111) was recently examined by Goyhenex [136] using RGL interactions. During molecular dynamics simulations for 1 nsec at 620 K he observed 18 rotations and 24 inter-cell motions. Translations in the same cell from fccfcc to hcp-hcp sites were also observed, but rarely. The potential energy changes in intracell diffusion are shown in Fig. 8.86a, and yielded a diffusion barrier of 0.26 eV for movement from hcp-hcp to fcc-hcp sites; movement continues to the stable fcc-fcc configuration over a barrier of 0.11 eV. For inter-cell motion the potential energy curve in Fig. 8.86b gave a barrier of 0.18 eV. The transition state is very broad and Goyhenex claimed that movement backward and forward from the transition state was possible, which limited the rate of success. Such a broad transition state is associated with a change in the dimer distance from 2.47 to 2.38 Å, due to the big mismatch in the sizes of Co and Pt. Molecular dynamics simulations yielded the Arrhenius plot of the diffusivity in

(a)

1.hcp-hcp

2.fcc-hcp

3.fcc-fcc

Co2/Pt

3.fcc-fcc

d(Co2) = 2.47 Å

d(Co2) = 2.24 Å

d(Co2) = 2.47 Å

2

0.25 0.20

2

0.15 0.10 0.05 0.00

2.intermediate (Co2 / Pt)

0.2 ΔE = E – Emin(eV)

ΔE = E – Emin(eV)

0.30

1.hcp-hcp

(b)

1

3

0.1

1

3

0.0

configurations

configurations

–14 Co2 on Pt(111)

–15 In D

–16 –17

(c)

–18 –19 ED = 0.25 ± 0.02 eV –5 2 –20 Do = 1.7 × 10 cm /sec 20

25

30 1/kT

Fig. 8.86

35

40

(eV–1)

Cobalt dimer diffusion on Pt(111) obtained in RGL calculations. (a) Potential energy change for Co2 moving in cell defined by six sites around substrate atom. (b) Potential energy curve for the inter-cell motion of Co2. (c) Arrhenius plot for diffusion of Co2 on Pt(111). Solid curve gives fit for dimer, dotted curve for monomer (after Goyhenex [136]).

646

Diffusivities of small clusters

Fig. 8.86c, from which an activation barrier of 0.25 ± 0.02 eV and a somewhat low prefactor of 1.7 × 10−5 cm2/sec were derived. The energy for dimer diffusion was slightly lower than for diffusion of the monomer. What is very surprising in this study is that intracell motion required more energy than inter-cell diffusion. The self-diffusion of clusters on the Pt(111) surface was widely investigated theoretically; unfortunately, the same cannot be said about hetero-diffusion.

8.13.5

Theoretical investigations: Pt(110) The leapfrog mechanism has been examined more closely for dimers and larger onedimensional clusters on Pt(110)-(1 × 2) surfaces by Montalenti and Ferrando [137], who resorted to RGL potentials. In molecular dynamics simulations of dimers, leapfrog transitions were observed most frequently at 1000 K. Much the same was found for trimers and tetramers. For dimers as well as trimers the energy for an atom to step up the {111} wall was 0.84 eV, while stepping down required 0.21 eV for the dimer, and 0.64 eV for the trimer. The crucial difference in leapfrog diffusion of dimers and trimers is caused by the existence of a metastable minimum for LF movement of the trimer, shown in Fig. 7.14; such a metastable minimum is not observed for movement of the LF dimer. The values obtained are close to the experimental estimate of 0.9 eV for up motion and 0.7 eV for down motion [123]. Estimates of platinum dimer diffusion along the channels of Pt(110)-(1 × 2) were made by Feibelman [138] in 2000 using the VASP [21,22] ab initio package. For diffusion by the leapfrog mechanism the barrier proved to be only 0.76 eV, significantly less than the estimate for diffusion of single Pt atoms. Movement on Pt(110)-(2 × 1) surfaces was examined with EAM potentials by Kürpick [42]. Four different diffusion mechanisms were considered and are indicated in Fig. 8.18. She took into account concerted motion of clusters, movement by the leapfrog mechanism as well as by exchange. It is clear that the leapfrog jump up is the limiting step for all the systems treated here. The potential energy experienced by Pt clusters is illustrated in Fig. 8.87. The concerted motion for trimers required a huge energy of 1.55 eV, tetramers needed 2.02 eV, and pentamers 2.48 eV. Exchange was also unlikely over a barrier of 1.30 eV for trimers and 1.36 eV for tetramers and pentamers. Step up in leapfrog motion

Energy (eV)

1.0 Pt3

0.8 0.6 0.4 0.2

Pt5

Pt4

0.0 0

4

8

12

0

4

8

12

0

4

8

12

Distance (Å) Fig. 8.87

Potential energy of one-dimensional platinum cluster diffusing by leapfrog mechanism on Pt(110)-(2 × 1) plane, calculated by using EAM potentials (after Kürpick [42]).

8.14 Clusters on gold surfaces

647

for chains three to five atoms long required 0.98 eV, a bit higher than estimated by Montalenti and Ferrando [137] and also higher than the experimental value. The values for stepping down, 0.70 eV for trimers and 0.67 eV for tetramers and pentamers, are in excellent agreement with experiment. She also looked at concerted movement on the unreconstructed surface, and for trimers came up with a barrier of 1.48 eV, tetramers 1.93 eV, and pentamers 2.39 eV. Recently Kim et al. [36] used molecular dynamics to look at the energetics of movement of linear clusters two to four atoms long on the unreconstructed Pt(110) surface. For dimers, in-channel hopping is the most probable step over a barrier of 0.842 eV, compared to 1.535 eV for exchange. For trimers, in-channel hopping required 1.195 eV and exchange 1.353 eV; for linear tetramers, however, exchange occurred over a barrier of 1.353 eV, while in-channel hopping was over 1.578 eV. For all of them, cross-channel hopping was a high-energy process. Unfortunately no experiments are available for diffusion on Pt(110) that would allow comparison with the calculated results.

8.14

Clusters on gold surfaces

8.14.1

Gold (100) Estimates of gold dimer mobility on the reconstructed Au(100) surface were done by Bönig et al. [139] drawing on EMT potentials for the interactions. Similar to the real system, in the simulation the top layer was reconstructed, but the second layer had the fcc(100) structure. This configuration caused two kinds of adsorption sites on the surface. They found a diffusion barrier of 0.2 eV for dimer movement, but motion of the dimer did not occur by a sequence of one-atom jumps. An important conclusion from this study was that jumps occurred with a bond length practically unchanged. Worth noting is that during adsorption of 2–6 atom clusters, local reconstruction to the fcc(100) structure of the lattice underneath the island was observed. Bogicevic et al. [13] used the VASP simulation package as well as the DACAPO code [14] to look at self-diffusion of Au dimers on the Au(100) unreconstructed plane. The two important processes were shearing, shown in the middle of the second row in Fig. 8.8 and to the left, which in fact can be achieved by a jump to the nearest-neighbor site perpendicular to the dimer axis. The second important process was stretching (middle row, center to right in Fig. 8.8), which can be achieved by a jump in the direction of the dimer axis. For Au2 as well as for Rh2 and Al2, jumping perpendicular to the dimer axis was a lower energy path than stretching with an energy of 0.86 eV compared to 1.11 eV. The authors argue that this is likely to be a general effect for other (100) fcc metals, but did not consider exchange in their investigations. Additionally, in real systems it is known that the Au(100) surface reconstructs. Molecular dynamics simulations of dimer diffusion have been carried out by Zhuang et al. [17] on the Au(100) surface using EAM potentials of Haftel et al. [71–73]. The simple exchange mechanism seems to be favorable for Au dimers, with an energy of 0.256 eV compared to 0.782 eV for traditional hopping. The existence of a complicated

648

Diffusivities of small clusters

exchange, as for example a 180 degree rotation-exchange, was also observed on this surface, and was associated with a stress-induced mechanism. Again reconstruction of the surface was not taken into account in this study. Investigation of dimer diffusion was continued by Liu et al. [19] with EAM [18] and RGL interactions [52,53]. The static activation energies from EAM potentials indicated that the exchange mechanism, with an energy of 0.2647 eV, was more likely than hopping, which occurred with an energy of 0.8151 eV. The difference in the activation energy calculated with RGL potentials for exchange and hopping was smaller – exchange required 0.3715 eV, hopping 0.5962 eV. However, both potentials show a preference for exchange on this plane. From 2 nsec molecular dynamics simulations with RGL potentials at 650 K, 33 exchange events were observed, but only 3 jumps and 3 complicated exchange mechanisms. Chang and Wei [20] used the VASP [21,22] package to establish the activation energy for Au dimers on Au(100). Just as in previous studies, they found exchange as the preferred mechanism of movement, with an energy of 0.24 eV, compared to 0.83 eV for hopping. Recently Kim et al. [36] used molecular dynamics to find the energetics of atom movement of linear clusters two to four atoms long, elongated along the [110] direction. For Au dimers, exchange required 0.404 eV, while perpendicular hopping demanded slightly more, 0.531 eV, and tangential hopping occurred over a barrier of 0.731 eV. For linear trimers and tetramers only tangential hopping in the [110] direction and exchange in the [100] were considered. Exchange for trimer atoms required 0.512 eV and for tetramer atoms 0.547 eV. Tangential hopping for trimer atoms took place over a barrier of 0.984 eV, and for tetramer atoms over 1.185 eV. On the Au(100) surface, movement of dimers has gotten most attention. The studies seem to indicate that exchange will be a leading mechanism of movement, and the scatter in activation energies obtained from different potentials is not too bad. After excluding data from RGL potentials, the value is always close to 0.25 eV for exchange and 0.80 eV for hopping, but experiments to allow a comparison are not available and in the real system the surface is known to reconstruct. This will complicate the movement of clusters and only Bönig et al. [139] looked at the movement of dimers on such a surface.

8.14.2

Gold (111) Diffusion on the (111) plane of gold has not been investigated in great detail – there are only three papers which briefly touch this subject. The first one by Chang et al. [27] used EAM potentials to calculate diffusion barriers for clusters of up to five atoms. For gold dimers they came up with a barrier of 0.10 eV, for trimers it was 0.25 eV, for tetramers the value was lower at 0.24 eV, but the barrier increased again for pentamers to 0.42 eV. The close-packed Au heptamer was investigated by Longo et al. [40] using EAM potentials parameterized according to Voter and Chen [41]. For a cluster gliding over its own crystal they came up with an energy of 0.64 eV. However in molecular dynamic observations they failed to observe gliding for Au7, which was attributed to the vertical mobility of the substrate at elevated temperatures. Recently Kim et al. [36] studied the energetics of dimer and trimer atoms moving. For dimers they found that the lowest-energy process was a jump in the direction 30 degrees to

649

8.14 Clusters on gold surfaces

the dimer axis at 0.100 eV; jumps at 90 degrees required twice as much energy, 0.229 eV, and exchange 1.484 eV. For triangular trimers jumping needed 0.218 eV, exchange 1.109 eV. No experiments have been done to provide an indication of what diffusion barriers for gold clusters should amount to. Additionally, all investigations were done on the unreconstructed surface, although it is known that the Au(111) surface reconstructs.

8.14.3

Gold (110) The leapfrog mechanism has been examined more closely for dimers and larger onedimensional clusters on Au(110) and Au(110)-(1 × 2) as well as platinum surfaces by Montalenti and Ferrando [137], who resorted to RGL potentials. In molecular dynamics simulations of dimers on Au(1 × 1) only concerted jumps of the dimer were observed, but for Au(110)-(1 × 2), leapfrog transitions were found most frequently, both at 450 K and 550 K. Much the same was observed for trimers, and tetramers. Concerted jumps of Au dimers on Au(110) required an energy of 0.47 eV, and 0.52 eV on Au(110)-(1 × 2). The energy of dimer atoms stepping up was estimated as 0.45 eV, while stepping down required only 0.14 eV, less than the concerted jump of a dimer. Trimer motion up requires the same energy as for dimers; stepping down, however, was more demanding, needing 0.37 eV, and much less than for the concerted jump of trimers, which was estimated at 0.7 eV. They also made estimates of the lifetimes of adatoms on top of the cluster and for gold concluded that the lifetime would be too short to be observed with dimers, but that it should be feasible to detect it by STM for trimers and higher. There still remains a question about how the atom from atop the cluster descends, whether this occurs by atom exchange or by an ordinary jump. It turned out that exchange events, both going up as well as going down, had a significantly larger activation energy than straight jumps; the exchange descent required 0.55 eV, and going up by exchange needed 0.64 eV. The jumps as dominant mechanisms in both descent and ascent were also confirmed by high-temperature simulations. The conclusion was that the leapfrog mechanism should be common in all sorts of fcc(110)-(1 × 2) surfaces. Diffusion of various dimers on Au(110)-(1 × 2) was further examined by Montalenti and Ferrando [140] using molecular dynamics with RGL interactions. Concerted jumps of the dimer, shown in Fig. 8.88a, as well as leapfrog diffusion, illustrated in Fig. 8.88b, were considered, but dimer dissociation was also allowed. Examined were the dimers Au2, Cu2, and AuCu; the barriers to the different atomic events they obtained are given in Table 8.14. For gold dimers, the energetics for leapfrog diffusion are displayed in Table 8.14 Activation barriers for leapfrog ELF, concerted jump ECJ and dissociation events Ed on as obtain by quenched molecular dynamics Au(110)-(1×2) [140]. Dimer

ELF(eV)

ECJ(eV)

Ed(eV)

Au2 Cu2 AuCu

0.45 0.75 0.58 (Au-LF) 0.67 (Cu-LF)

0.52 0.66 0.60

0.51 0.50 0.52 (Au-Di) 0.54 (Cu-Di)

650

Diffusivities of small clusters

Fig. 8.88

Dimer diffusion on Au(110)-(1 × 2) examined with RGL potentials. (a) Schematic of concerted jump of dimer on Au(110)-(1 × 2). (b) Model for leapfrog mechanism in dimer diffusion on Au(110)-(1 × 2). (c)–(f) Schematic of energy changes in dimer diffusing over Au(110)-(1 × 2). (c) Au2, (d) Cu2, (e) AuCu with Cu leapfrog path, (f) AuCu with Au leapfrog path. T1 denotes saddle point for diffusion (after Montalenti and Ferrando [140]).

Fig. 8.88c. What is shown here is not a universal curve. For Au2, the first saddle point constitutes the maximum, and therefore corresponds to the activation energy for diffusion. Leapfrog motion plays a major role. For Cu2 dimers the energy is configured differently, shown in Fig 8.88d, in that the peak lies not at the first barrier but rather at the second, and this is also the situation for AuCu dimers, as is clear in Fig. 8.88e–f. For all three the barrier to dissociation (in Table 8.14) is higher than for single atom diffusion, so that if dissociation takes place the adatoms will move rapidly and recombine with each other. It should be noted that for Cu2 the barrier to leapfrog motion is higher than for concerted jumps, which therefore dominate. Au dimers are also more stable than Cu dimers. However, for all three dimers, leapfrog diffusion does occur, and this seems to be a general trend on (110)-(1 × 2) surfaces. Diffusion of nickel atoms and clusters has been simulated at 400 K on the Au(110)(1 × 2) plane by Fan and Gong [141] using Johnson potentials [18]. Single adatom motion on the reconstructed (110) has already been described. Dimer motion is somewhat similar, as illustrated in Fig. 8.89a. Motion can occur by concerted jumps along the trough with the dimer bonds parallel to the trough, or else with one atom in the trough, the other along the sidewall. Diffusion may also take place with both atoms on the sidewall, or by jumping from one channel to a neighboring one. The last is not a frequent occurrence, however. Occasionally leapfrog diffusion may take place, with one atom jumping to the sidewall and then to the other side of the first atom in the trough, but this is not as frequent as concerted jumping. The movement of Ni dimers on Au(110)-(1 × 2) is definitely different than movement of Cu or Au dimers. Nickel trimers are in a triangular configuration with two atoms at the bottom of the trough and the third near the local minimum on the sidewall. Some of the diffusion steps are illustrated in Fig. 8.89b. Diffusion may involve concerted jumps

8.15 Comparisons

Fig. 8.89

651

Trajectories for nickel clusters diffusing on Au(110)-(1 × 2) relying on Johnson potentials. (a1) Concerted jumps in trough. (a2) One atom moves in trough, the other one on the side. (a3) Dimer diffusion on side wall. (a4) Jump from one trough to the next. (b) Diffusion of Ni3 on Au(110)-(1 × 2). (b1) Translation in trough. (b2) Rotation. (b3) Translation plus rotation. (b4) Jump over trough by rotation and translation (after Fan and Gong [141]).

along the trough, similar to gliding (Fig. 8.89b1), rotation (Fig. 8.89b2), translation plus rotation (Fig. 8.89b3), and inter-channel motion by rotation and translation (Fig. 8.89b4). Compared to gliding, rotation had quite a low energy because only two atoms move instead of three. Higher clusters were three-dimensional, and moved rapidly by rolling over the surface, but such structures are not considered here. The NiN cluster had a sphere-like shape due to stronger Ni-Ni compared with Ni-Au interactions. The work showed that the diffusion mechanism depended on the size of the cluster, but no attention was paid to the energetics of movement. These novel mechanisms of movement should still be confirmed experimentally. However, they showed the importance of in-cluster interactions on diffusion. Kim et al. [36] looked at jumps of linear Au clusters two to four atoms long on an unreconstructed Au(110). For dimers, in-channel jumps were most likely, with an energy of 0.456 eV, exchange required 0.906 eV, and cross-channel jumps 1.187 eV. For trimers, in-channel jumps needed 0.646 eV, exchange 0.756 eV, and cross-channel jumps 1.763 eV. For tetramers, in-channel jumps occurred over a barrier of 0.847 eV, exchange over 0.989 eV, and cross-channel jumps over 2.902 eV.

8.15

Comparisons Much material has been surveyed here. There is a question that still remains: are there any significant features to indicate trends in the diffusion behavior? Here it is important to notice that considerable work on cluster diffusion has been done on the (110)

652

Diffusivities of small clusters

plane of tungsten. For nickel, palladium, platinum, and iridium, clusters of a few atoms all exist as linear chains oriented along . The activation energies for diffusion rises starting from the lowest value for palladium and going to nickel, and then platinum (see Table 8.10). Presumably the barrier for diffusion of iridium clusters is higher still. The fact that palladium is lowest may be surprising, as this is not in keeping with the trends of the enthalpy of vaporization. However, we should have expected this, as the diffusion of palladium and nickel single adatoms on W(211) behaves in the same way. Rhenium and tantalum clusters are different, and what is interesting is the formation of Re dimers. On the W(110) surface, Re dimers are unstable, but at the same time both in-channel and cross-channel forms are stable on the W(211) surface. The question immediately arises if stability of Re2 is surface mediated, and is favorable for W(211) and not favorable on the W(110) structure, or if the W(110) structure provides unfavorable bond distances for Re2 to exist. It will be important to probe this question on different bcc surfaces. What little has been done on rhenium shows trimers both linear and triangular, with bigger clusters twodimensional. The energetics of diffusion have not been established quantitatively, but mobility increases significantly for tetramers and pentamers, which is probably correlated with the intriguing open structure of these clusters. Palladium clusters have been examined on Ta(110), another bcc metal. Just as on W(110), these clusters exist in the form of chains, but with nine palladium atoms, two-dimensional structures form again, so diffusion should proceed similarly to chains on W(110). Clusters have been studied in experiments on the close-packed surfaces Pt(111), Ir(111), as well as Re(0001). Iridium and platinum are fcc neighbors in the periodic table, but cluster behavior is not too similar. With iridium, there is a marked drop in the diffusion barrier of iridium tetramers compared to trimers, but that is not seen with platinum clusters on Pt(111). For monomers, the self-diffusion barrier of iridium is higher than for platinum, as expected from the enthalpy of vaporization. However, this relation is not maintained uniformly in going to clusters with a larger number of atoms. Oddly enough, the situation improves in comparing the (111) plane of fcc iridium with hcp Re(0001). As shown in the plot in Fig. 8.90, the barriers for self-diffusion of rhenium clusters up to Re7 are uniformly higher, but both tetramers have a minimum value. Furthermore, the behavior of rhenium compared to iridium is precisely what is expected from the enthalpies of vaporization of the two materials. What seems to be a common feature of clusters on Ir(111) and Pt(111) is that heptamers are very stable, and have the possibility of gliding over the surface without shape change. However, according to molecular dynamics studies, gliding of heptamers is not observed on all fcc(111) surfaces [40]; this situation should be checked in experiments. For movement and shearing on fcc(111) surfaces, two concerted mechanisms were predicted by simulations – reptation and dislocation motion; however, these should still be probed directly in experiments. We now turn to clusters on the (100) planes of fcc elements. On platinum (100), dimer diffusion occurs by exchange, trimers seem to move by exchange plus hopping but the clusters are linear. On Rh(100), clusters tend to be two-dimensional, and move with an apparently oscillating activation energy, by atom hopping over the surface. Diffusion of platinum clusters on the (100) plane of rhodium occurs by hopping, but the clusters are

653

8.15 Comparisons

Table 8.15 Barriers to dimer self-diffusion (eV). Material

Experiment

Pt(111) Ir(111) Cu(111) local Pt(100)

0.37 ± 0.02 [121] 0.447 ± 0.013 [120] 0.018 ± 0.003 [44] 0.41 [33]

Ir(100)

0.88 ± 0.05 [108]

Calculation 0.35 DFT [130] 0.467 RGL [114] 0.009 EAM [58] 0.52 EAM [33] 0.644 EAM [19] 1.078 RGL [114]

0.17 EAM [27] 0.24 EAM [27] 0.016 EAM [55] 0.18 VASP [20] 0.292 SEAM [19] 0.65 EAM [115]

0.16 EMT [122]

0.372 EAM [17] 0.799 MD [36] 1.30 VASP [20]

1.6 IrN on Ir(111)

1.4

Activation Energy ED (eV)

ReN on Re(0001) 1.2

1

0.8

0.6

0.4

0.2

Fig. 8.90

0

1

2

3 4 5 Size of Cluster (atom)

6

7

8

Comparison of self-diffusion barriers observed in FIM for clusters on Ir(111) and Re(0001).

again in the form of linear chains, indicating the importance of long-range interactions. Predictions or comparisons are not easy here. At least three competing mechanisms are involved in the movement on fcc(100) planes – exchange, individual atom hopping and shearing. However, on fcc(100), concerted motion of the whole cluster was not observed. Diffusion of iridium clusters on Ir(100) appears to be different, in that clusters usually appear as either linear chains or two-dimensional islands, the structure depending upon the temperature; low temperatures for linear chains, but at higher temperatures rearrangement to island form occurs. In any event, only little is known about surface diffusion. In special cases, when clusters are arranged in linear form, diffusion steps can be crudely estimated, but in general, with present knowledge, prediction about the surface diffusion of clusters is difficult. With the various computational efforts describing cluster motion in place, the question arises – how useful are they. One thing fairly clear is that the calculated barrier heights are not uniformly reliable. That emerges on looking at Table 8.15, where values determined

654

Diffusivities of small clusters

Table 8.16 Comparison of self-diffusion barriers from experiment and calculation [27] on (111) planes for Pt [121] and Ir [29]. ED (eV) for Pt Number of Atoms

Expt

EAM

Expt

EAM

1 2 3 4 5 6 7

0.260 0.37 0.52 0.57 0.78 0.89 1.17

0.07 0.17 0.40 0.39 0.61

0.290 0.45 0.65 0.48 0.69 ~0.92 1.49

0.11 0.24 0.56 0.54 0.83

(b) 1.2 Activation Energy ED (eV)

Activation Energy ED (eV)

(a) 1.5 Experiment Calculations

1

0.5 IrN on Ir(111) 0

Fig. 8.91

ED (eV) for Ir

0

1

2 3 4 5 6 7 Size of Cluster (atom)

8

Experiment Calculations

1 0.8 0.6 0.4

PtN on Pt(111)

0.2 0

0

1

2 3 4 5 6 7 Size of Cluster (atom)

8

Comparison of diffusion energies for clusters calculated using EAM potentials [27] with experimental values for iridium clusters on Ir(111) [26,120] and platinum clusters on Pt(111) [121].

in direct observations of dimers using the FIM are compared with calculations of different kinds for the same materials. On fcc(111) planes some of the calculational results are in excellent agreement with the experiments, but some are not. For dimers on fcc(100) planes, the calculated diffusion barriers differ quite significantly from the experiments. It must be emphasized, however, that although the calculations offer a somewhat uncertain guide to absolute barrier heights, they are nevertheless very useful in comparisons of how size affects cluster behavior. This emerges from Table 8.16, as well as from Fig. 8.91, where calculations are compared with experiments for both Pt(111) and Ir(111). The trends in the former are fairly well duplicated by the theoretical efforts. Most important have been the calculations dealing with atomic jumps in surface diffusion, which are very difficult to derive in experiments. These have made it clear that diffusion of clusters in some instances does not occur by independent single jumps, but proceed by the correlated motion of more than one atom at a time, sometimes by separate motion of cluster parts, that is by dimer and trimer shearing. Calculations have also served to unravel how leapfrog movement of clusters takes place, and have clarified

655

8.15 Comparisons

conditions under which atom exchange facilitates cluster motion. It is in this area of understanding the mechanisms of diffusive processes that calculations have and will probably make the greatest contributions. It also is worth noting the terrible lack of theoretical efforts to describe diffusion on bcc surfaces, especially for tungsten, where experimental effort has been concentrated. A survey of experimentally determined energetics and prefactors for cluster diffusion is provided in Table 8.17.

Table 8.17 Experimentally measured movement of the center of mass of clusters ν0 (sec−1)

Method

Ref.

1 × 1012 ± 0.5 8 × 1011 ± 0.5

STM STM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM STM STM FIM FIM

[142] [44] [46] [46] [46] [46] [46] [46] [46] [46] [46] [46] [46] [65] [65] [65] [65] [65] [65] [57] [57] [82] [82]

1.8(×5.1±1) × 10−3

FIM

[82]

0.79 ± 0.02

5.2(×2.3±1) × 10−4

FIM

[84]

0.90 ± 0.07 0.92 ± 0.14 0.67 ± 0.06 0.67 ± 0.06 0.79 ± 0.16 0.87 ± 0.16 0.91 eV ~1.22 ~1.02

6.2(×13±1) × 10−3 1.4(×160±1) × 10−3 30.6(×20.1±1) × 10−8 9.21(×20.1±1) × 10−8 15.2(×403±1) × 10−8 45.6(×403±1) × 10−8 † −3 10 † −3 10 † −3 10

FIM FIM FIM FIM FIM FIM FIM FIM FIM

[87] [87] [90] [90] [90] [90] [143] [93] [93]

System

Mechanism

ED (eV)

Cu/Cu(111) Cu2/Cu(111) Rh/Rh(100) Rh2/Rh(100) Rh3/Rh(100) Rh4/Rh(100) Rh5/Rh(100) Rh6/Rh(100) Rh7/Rh(100) Rh8/Rh(100) Rh9/Rh(100) Rh10/Rh(100) Rh12/Rh(100) Pt/Rh(100) Pt2/Rh(100) Pt3/Rh(100) Pt4/Rh(100) Pt5/Rh(100) Pt6/Rh(100) Cu/Ag(111) Cu2/Ag(111) Re/W(211) Re2/W(211) crosschannel Re2/W(211) in-channel Re3/W(211) crosschannel W/W(110) W2/W(110) Pt/W(110) Pt2/W(110) Pt3/W(110) Pt4/W(110) Re/W(110) Re3/W(110) Re5/W(110)

Jump Jump local Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump

0.040 ± 0.001 0.018 ± 0.003 0.84 0.97 1.02 1.05 1.01 1.14 1.04 1.14 1.22 1.07 1.23 0.92 ± 0.13 0.91 1.02 1.03 1.03 1.16 0.065 0.073 0.86 ± 0.030 0.78 ± 0.013

Jump

1.01 ± 0.05

Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump

Do(cm2/sec)

†

10−3 10−3 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 2 × 10−3 ± 1.9 † 1 × 10−3 † 1 × 10−3 † 1 × 10−3 † 1 × 10−3 † 1 × 10−3 †

2.2(×2.8±1) × 10−3 4.5(×1.7±1) × 10−4

1 × 1012 ± 0.5 † 12 10

656

Diffusivities of small clusters

Table 8.17 (cont.) System

Mechanism

ED (eV)

Pd-Re/W(110) Pd/W(110) Pd2/W(110) Pd3/W(110) Pd4/W(110) Pd5/W(110) Pd6/W(110) Pd7/W(110) Pd8/W(110) Pd9/W(110) W/W(111) W2/W(111) W3/W(111) W4/W(111) W5/W(111) W6/W(111) W7/W(111) W8/W(111) W9/W(111) W10/W(111) Re/Re(0001) Re2/Re(0001) Re3/Re(0001) Re4/Re(0001) Re5/Re(0001) Re6/Re(0001) Re7/Re(0001) Re8/Re(0001) Ir/Ir(100) Ir2/Ir(100) Ir4/Ir(100) Rh/Ir(100) Rh2/Ir(100) Rh4/Ir(100) Re-Ir/Ir(100)αRe Re-Ir/Ir(100)βRe Ir/Ir(111) Ir2/Ir(111) Ir3/Ir(111) Ir4/Ir(111) Ir5/Ir(111) Ir6/Ir(111) Ir7/Ir(111) Ir18/Ir(111) Ir19/Ir(111) Ir/Ir(110) Ir/Ir(110) Ir2/Ir(110)

Jump local Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Ex Ex Jump Jump Jump Jump Jump local Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Jump Ex Jump

0.48 ± 0.02 0.51 0.65 0.68 0.75 0.80 0.84 0.88 0.88 0.95 1.9 ~1.62 ~1.74 ~1.79 ~1.7 ~1.74 ~1.81 ~1.83 ~1.9 ~2.01 0.48 ± 0.02 ~0.43 ~0.65 ~0.43 ~0.89 ~1.12 ~1.21 ~1.25 0.84 ± 0.05 0.88 ± 0.05 1.09 0.80 ± 0.08 0.83 ± 0.06 0.93 0.60 ± 0.07 0.73 ± 0.07 0.29 ± 0.003 0.45 ± 0.01 0.65 ± 0.02 0.48 ± 0.01 0.69 ± 0.02 ~0.92 1.49 ± 0.03 1.63 ± 0.07 2.54 ± 0.06 0.80 ± 0.04 0.71 ± 0.02 1.05 ± 0.14

Do(cm2/sec)

ν0 (sec−1)

†

10−3 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 6.13(×2.6±1) × 10−6 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 † −3 10 6.26(×11±1) × 10−2 1.42 × 10−3 ± 0.8 2.09 × 10−3 ± 1.5 3.53 × 10−4 ± 1.1 4 × 10−4 ± 1.5 2 × 10−2 ± 1.5 3.8(×1.4±1) × 10−4 2.6(×2.2±1) × 10−5 4.2(×2.4±1) × 10−4 1.5(×2.3±1) × 10−5 6.0(×1.9±1) × 10−6 ~1 × 10−5 1.4(×2.4±1) 7.8(×4.5±1) × 10−4 13(×2.6±1) 4 × 10−3 ± 0.8 6 × 10−2 ± 1.8 3.7 × 10−5.0 ± 1.3

2.0 × 1012.0 ± 1.5 2.6 × 1013.0 ± 1.5

Method

Ref.

FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM

[94] [99] [99] [99] [99] [99] [99] [99] [99] [99] [100] [100] [100] [100] [100] [100] [100] [100] [100] [100] [103] [103] [103] [103] [103] [103] [103] [103] [106] [108] [109] [108] [108] [109] [107] [107] [29] [29] [29] [29] [29] [29] [29] [111] [111] [144] [144] [113]

657

References

Table 8.17 (cont.) System

Mechanism

ED (eV)

Do(cm2/sec)

Ir2/Ir(110) Pt/Pt(100) Pt2/Pt(100) Pt3/Pt(100) Pt/Pt(111) Pt2/Pt(111) Pt3/Pt(111) Pt4/Pt(111) Pt5/Pt(111) Pt6/Pt(111) Pt7/Pt(111)

Ex Ex Ex Ex+Jump Jump Jump Jump Jump Jump Jump Jump

1.18 ± 0.12 0.47 0.41 0.49 0.260 ± 0.003 0.37 ± 0.02 0.52 ± 0.01 0.57 ± 0.04 0.78 ± 0.02 0.89 ± 0.04 1.17 ± 0.04

2.6 × 10−4.0 ± 0.7 1.3(×10±1) × 10−3

†

2.0(×1.4±1) × 10−3 1.9(×4.5±1) × 10−4 1.1(×2.1±1) × 10−3 6.6(×6.1±1) × 10−5 1.8(×2.3±1) × 10−2 4.9(×5.1±1) × 10−3 5.1(×3.8±1) × 10−1

ν0 (sec−1)

Method

Ref.

FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM FIM

[113] [145] [33] [33] [121] [121] [121] [121] [121] [121] [121]

assumed value

References [1] D. W. Bassett, M. J. Parsley, Field ion microscope observations of cluster formation in metal deposits on tungsten surfaces, Nature 221 (1969) 1046. [2] D. W. Bassett, Controlled clustering: a technique for use in studying thin film nucleation in the field ion microscope, J. Phys. E: Sci. Instr. 3 (1970) 417–418. [3] D. W. Bassett, Field ion microscope studies of iridium adatom clusters on (110) tungsten surfaces, Surf. Sci. 21 (1970) 181–185. [4] D. W. Bassett, The use of field ion microscopy in studies of the vapour deposition of metals, Surf. Sci. 23 (1970) 240–258. [5] T. T. Tsong, Interaction of individual metal atoms on tungsten surfaces, J. Chem. Phys. 55 (1971) 4658–4659. [6] T. T. Tsong, Direct observation of interactions between individual atoms on tungsten surfaces, Phys. Rev. B 6 (1972) 417–426. [7] D. W. Bassett, D. R. Tice, The stability of W2 and WRe adatom clusters on (110) tungsten surfaces, Surf. Sci. 40 (1973) 499–511. [8] J. C. Tully, G. H. Gilmer, M. Shugard, Molecular dynamics of surface diffusion. I. The motion of adatoms and clusters, J. Chem. Phys. 71 (1979) 1630–1642. [9] D. Ghaleb, Diffusion of adatom dimers on (111) surface of face centered crystals: A molecular dynamics study, Surf. Sci. 137 (1984) L103–108. [10] P. J. Feibelman, Rebonding effects in separation and surface-diffusion barrier energies of an adatom pair, Phys. Rev. Lett. 58 (1987) 2766–2769. [11] C.-L. Liu, J. B. Adams, Structure and diffusion of clusters on Ni surfaces, Surf. Sci. 268 (1992) 73–86. [12] S. Valkealahti, M. Manninen, Diffusion on aluminum-cluster surfaces and the cluster growth, Phys. Rev. B 57 (1998) 15533–15540. [13] A. Bogicevic, S. Ovesson, B. I. Lundqist, D. R. Jennison, Atom-by-atom and concerted hopping of atom pairs on an open metal surface, Phys. Rev. B 61 (2000) R2456–2459. [14] B. Hammer, Bond activation at monatomic steps: NO dissociation at corrugated Ru(0001), Phys. Rev. Lett. 83 (1999) 3681–3684.

658

Diffusivities of small clusters

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9

Diffusion of large clusters

Starting in the 1970s, considerable work was done on dimers and trimers and their surface diffusion, but there were no experimental studies of larger clusters, containing twenty or more atoms, since they were assumed to be immobile at the surface. This changed in 1984, with the work of Fink [1–3] using the FIM, in which he assembled a cluster of twenty or more palladium atoms on the (110) plane of tungsten. At 390 K, this large cluster moved over the surface as a unit, as shown in Fig. 9.1, demonstrating its diffusivity. Large clusters turn out to be mobile at relatively low temperatures and their movement needed to be investigated, since it influences the stability of nanostructures and thin film growth kinetics. With the invention of the scanning tunneling microscope, large clusters were rediscovered a few years later, and work began to unravel how diffusion occurred, many of the studies focusing on the dependence of diffusivity on cluster size. This effort will be surveyed, arranged according to the type of the surface. Study of large clusters began with the examination of movement on a bcc surface, on W(110), but this work was not continued later; instead fcc surfaces were investigated in detail.

9.1

Large clusters on fcc(100) surfaces Theoretical investigations of large clusters on fcc(100) surfaces started in 1980 with the work of Binder and Kalos [4], which initiated a number of discussions of how the cluster diffusivity D was affected by the size and the specific mechanism of diffusion. Binder and Kalos had already derived simple estimates for clusters of radius Rr in a square lattice gas and showed tentatively that the dependence of the diffusivity on cluster size could be explained in terms of a crossover between two mechanisms – peripheral movement and evaporation–condensation, mechanisms yielding different power laws. For small clusters they expected the diffusivity to depend on the number of atoms N in the cluster to the power 1 + 1/d; for larger clusters the exponent would be 1 − 1/d, where d is the dimensionality; the location of the crossover depended on the temperature. The theoretical investigations were continued with Rh clusters on the Rh(100) surface evaluated with Lennard-Jones potentials by Voter [5]. He also observed a crossover in diffusivity between small and large clusters, as is shown in Fig. 9.2a. For clusters larger than 10 atoms the exponent in the size dependence of the diffusivity was 1.76 ± 0.05 and the main mechanism was shown to be edge running. This conclusion was based on

665

9.1 Large clusters on fcc(100) surfaces

Fig. 9.1

Formation and diffusion of Pd cluster on W(110) plane [3]. (a) Clean surface. (b) Roughly twenty Pd atoms have been deposited. (c) Pd chains have formed after heating to 235 K. (d) After additional deposition and heating, 2D clusters have formed. (e) At 340 K, a large Pd cluster has been created. (f) Cluster has moved over the surface at 390 K. 2.10 eV

(a)

(b) –10

RhN/Rh(100) T = 2000 K

1 2

In D

3

–15

1.38 eV

4 5

0.68 eV 0.61 eV

6 7 10 8

–20

15 12

20

αN =

–25

1.35 eV

30

1.76

75

± 0.

05 0.85 eV

1

2

3

0.69 eV

50

0.80 eV

4

In N Fig. 9.2

Diffusion of rhodium clusters on Rh(100) obtained from Monte Carlo simulations using Lennard-Jones potentials. (a) Size dependence of Rh cluster diffusivity. (b) Energetics for movement of atom onto a fresh cluster edge (after Voter [5]).

agreement between the energy required for movement of 20 atom clusters (2.08 ± 0.04 eV) and the energy for an atom climbing up to the edge from a kink site (2.10 eV), which is the limiting step for adatom edge running. This as well as other energies calculated for adatom edge running are shown in Fig. 9.2b. A year later Voter [6] investigated large clusters on Ag(100). The crossover between the diffusivity of small and large clusters was not so clear for this system, as indicated in Fig. 9.3a. However, similar to the previous study, the activation energy for movement of a 100 atom cluster, 0.82 ± 0.09eV, was in very good agreement with the energy required for the step that limits peripheral movement – an adatom jumping out from a kink site, 0.83 eV. The energetics of other steps are shown in Fig. 9.3b. Experimental studies on fcc(100) surfaces started with an extensive piece of work by Wen et al. [7] dealing with the diffusion of large silver clusters on the Ag(100) surface.

666

Diffusion of large clusters

(a)

0.83 eV

(b)

–15

0.50 eV

Ag/Ag(100) T = 500 K –20 In D

0.25 eV 0.22 eV

–25 0.55 eV

–30 –35

0.22 eV

adatom cluster diffusion vacancy cluster diffusion 0.25 eV

0

1

2

3

4

0.26 eV

5

In N Fig. 9.3

Silver cluster diffusion on Ag(100) surface. (a) Dependence of cluster diffusivity on size. (b) Energetics of atom movement to fresh cluster edge (after Voter [6]).

log10D

–17

–18

–19 0

Fig. 9.4

200

400 N

600

800

STM observations of room temperature diffusivity of Ag cluster on Ag(100) as a function of the number of cluster atoms N (after Wen et al. [7]).

The clusters were made up of from 100 to 800 atoms, and were observed at room temperature with the STM. A plot of the diffusivity against the number of cluster atoms is shown in Fig. 9.4. Wen et al. considered two different mechanisms for the motion of the clusters: atom movement around the periphery of the cluster, and evaporation– condensation, in which the cluster is in contact with a dilute two-dimensional gas layer. For peripheral diffusion, Monte Carlo simulations had shown the diffusivity to depend upon the number of cluster atoms N, with D ¼ D N
αN ;

(9:1)

here αN lies between 1.5 and 2. These results are shown by the solid line in Fig. 9.4, which deviates significantly from the experiments. For the evaporation–condensation mechanism of cluster diffusion they estimated that αN would be 0.5. This mild dependence upon the number of cluster atoms is given by the dotted line in the figure, and seems to better agree with the experimental data. Direct evidence for evaporation of atoms from the cluster edges was found from observations of the disappearance of small clusters. In short, the mechanism for cluster diffusion was attributed to the evaporation and condensation of cluster atoms.

667

9.1 Large clusters on fcc(100) surfaces

8.3

11.4

LL (atoms) 15.8 21.8 8.3 13.2

20.9

log D (D:Å2/sec)

10–1

(a) Copper αL = 2.49±0.09 0.92

Fig. 9.5

33.1

1.06

10–2

(b) Silver αL = 2.28±0.10

1.12 1.20 1.34 0.92 log (LL) (LL:atoms)

1.32

1.52

Diffusivity of islands measured with STM in its dependence upon island length LL ( after Pai et al. [8]).

These findings almost immediately stimulated controversy. Pai et al. [8] came out with an STM examination of the diffusivity and coarsening of silver on Ag(100) as well as copper on Cu(100). In the relation between diffusivity and the island length LL they found L D ¼ D L
α L

(9:2)

with αL = 2.28 ± 0.10 and D* = (18.5 ± 4.5) Å2/sec for silver and 2.49 ± 0.09 and D*= (59.5 ± 13.1) Å2/sec for copper, as shown in Fig. 9.5, for islands of more than a thousand atoms. These coefficients are reasonably close to what is expected for correlated evaporation–condensation or terrace limited diffusion. However, separate observations revealed essentially no decay of the islands. For copper, hardly any loss was noted during a period of six hours. With silver, the decay rate led to an evaporation–condensation diffusivity of  2.2 × 10–5 Å2/sec. These values are three orders smaller than the measured diffusivity of D  (3–6) × 10–2 Å2/sec. The conclusion was that movement of atoms around the cluster periphery was responsible for the diffusivity. The authors claimed that the usual assumption of a structureless periphery was responsible for such disagreement, and to deduce the macroscopic mechanism details of the structure had to be taken into account. They also performed an analysis with a constant kink density, and random attachment and detachment of atoms from kinks. They found that αL changed from 1 to 3 with increasing radius of the clusters. This study was based on much better statistics than that of Wen et al. [7] and challenged their findings. Wen et al. [9], in their later study on the coarsening mechanism on Ag(100), observed no Ostwald ripening of islands up to a coverage of 0.65 ML, which might indicate the same problem. However they rationalized the lack of Ostwald ripening by a huge probability of recondensation close to the cluster. Coarsening of large silver clusters on a silver(100) surface was also measured with a scanning tunneling microscope (STM) at 295 K by Stoldt et al. [10] and the Ostwald

668

Diffusion of large clusters

4.5

1.1 1.0

Nav /Nav 0

0.9 Sav0 = 140 atoms

0.8

Sav0 = 60 atoms

0.7 0.6 0.5 0.4

Fig. 9.6

(a) 0

50

Sav0 = 40 atoms

100 150 200 250 300 350 400 Time (min)

Nav (×10–5Å–2)

Sav0 = 270 atoms

4.0

3.5 (b) 3.0

0

50 100 150 200 250 300 350 400 Time (min)

Coarsening kinetics for Ag islands on Ag(100) determined with STM. (a) The unstrained surface, fitted with Do = 50 Å2/sec and an exponent of 1.5. (b) The strained surface, fitted with Do = 250 Å2/ sec and exponent of 1.5 in Eq. (10.1). Dashed curve for Do = 50 Å2/sec (after Stoldt et al. [10]).

ripening process was comparable with experimental noise in this study. They found the coarsening rate decreased strongly with increasing size of the islands (coverage), suggesting a strong decrease in the diffusivity with increasing island size. The results of their experiments are shown in Fig. 9.6a, where the data were analyzed based on mean field theory and were fitted with a value of 50 Å2/sec for D* and an exponent αN = 1.5 in Eq. (9.1), the value appropriate for perimeter diffusion of atoms around the clusters. They also observed enhanced diffusion rates for islands on a strained Ag(100) surface; their data are shown in Fig. 9.6b, where their findings are fitted with 250 Å2/sec for D* and keeping an exponent αN = 1.5 in Eq. (9.1). The origin of the strain was undetermined. Monte Carlo simulations were carried out by Pal and Fichthorn [11] for the diffusion of large clusters on an fcc(100) surface at 300 K. In Monte Carlo simulations they used a lattice model with periodic boundary conditions and nearest-neighbor interactions on a square lattice. An activation energy of 0.52 eV was assumed for single atom movement, together with a lateral bond strength of 0.8 eV, and a frequency factor of 1012 sec−1. Individual atomic steps in cluster diffusion were monitored. Four elementary processes are shown in Fig. 9.7a: atomic diffusion by uncorrelated jumps around the cluster periphery PD, the hopping of a kink atom KD, the attachment KA of an isolated atom to a kink position, and finally the process designated as CPD, a variant of peripheral diffusion, in which an atom jumps from a site where it makes two bonds, to an adjacent position, where it again makes two bonds. The plot for the diffusivity as a function of size is given in Fig. 9.7b, and divides into two sections, the first with a slope of 1.47 for clusters made of fewer than 100 atoms, and the second with a slope of 0.49 for large clusters of more than 100 atoms. For small clusters, diffusion around the edges of the clusters proved to be the dominant step, but for large clusters movement of kink atoms turned out to be the most frequent event, shown in Fig. 9.7c. Evaporation–condensation (EC) suggested from the scaling factor was not found responsible for the observed value of αN  0.5. It was instead the increase in KA/KD events, together with some rise in the CPD process, that was important for the behavior of the cluster. Some oscillations in the

669

9.1 Large clusters on fcc(100) surfaces

1.00

D/a2 sec

slope = –1.47

(a)

0.10

CPD

dd

KA

slope = –0.49

(b) 0.01 101

102 103 Number of cluster atoms, N

100 (c)

PD

KD

Percent

80 60 40

mechanism EC CEC PD KD CPD

20 0 20 Fig. 9.7

100 1000 2000 Number of cluster atoms, N

Diffusion on fcc(100) surfaces. (a) Schematic of atom movements around cluster periphery. Empty circles indicate positions that can be filled by adatom. KD – jump of kink atom. PD – peripheral diffusion. KA – kink attachment of isolated adatom. CPD – two bonds broken, two formed. (b) Dependence of diffusivity of clusters on N, the number of cluster atoms. (c) Probability of occurrence for different mechanisms. EC – evaporation–condensation, CEC – evaporation and condensation within two hops (after Pal and Fichthorn [11]).

diffusivity of large clusters were also observed. From a separate analysis of the mechanisms involved in movement, shown in Fig. 9.7c, the condensation–evaporation process was found very rarely, and if an atom detached from the cluster, then reattachment usually was observed within two hops, a process named CEC, evaporation and condensation spatially correlated, in Fig. 9.7c. Extensive Monte Carlo simulations of diffusion and evaporation for large clusters on fcc(100) surfaces were also done by Lo and Skodje [12], based on a bond breaking model. They allowed for hops only to nearest-neighbor sites, and neglected any concerted gliding, adatom/substrate exchange processes, or dimer shearing. Clusters with as many as 100 000 atoms were examined, at a temperature around 500 K, where both large clusters and monomers were stable. The behavior of the diffusivity as a function of the number N of atoms in the cluster is plotted in Fig. 9.8a. Examination of the diffusion revealed that the primary event was correlated evaporation–condensation, in which an evaporating atom reattaches close to the same site, in agreement with the value found for the αN exponent. From an Arrhenius plot of the diffusivity in Fig. 9.8b, an activation energy was derived, in agreement with the energy for the removal of an atom from a kink site, shown in the inset. The same value was obtained from the evaporation rate of islands as large as 10 000 atoms. Based on this observation, the authors concluded that removing

670

Diffusion of large clusters

105

106

D205(a2 /sec)

D(a2 /sec)

105 104

(b)

104

103 (a) 102

10

103 100 1000 Island Size, N

10000

0.8

0.9

1.0 1.1 1/kT

1.2

1.3

D(a2 /sec)

105

104

103 (c) 102

Fig. 9.8

100

1000 Island Size, N

10000

Surface diffusivities on fcc(100) derived in MC simulations. (a) Dependence of cluster diffusivity upon the number of cluster atoms N. Solid line has slope of 1.01 ± 0.05, a‘ is the lattice spacing. (b) Arrhenius plot for cluster of 205 ± 10 atoms with an energy of 6.05 ± 0.08 ES. ES is the smallest barrier in the system. (c) Dependence of vacancy island diffusivity upon the number of missing cluster atoms; upper plot without substrate, lower with substrate (after Lo and Skodje [12]).

atoms from kink sites was the limiting step in the kinetics of island movement. Here we have to emphasize the different dependence on cluster size from that found by Pal and Fichthorn [11], obtained at lower temperatures and for a smaller range of cluster sizes. The atomic mechanism in the diffusion was also different: for Pal and Fichthorn it was kink movement, while for Lo and Skodje correlated evaporation–condensation. For vacancy islands the dominant mechanism was not derived directly, but from the exponential dependence of the diffusivity on island size in Fig. 9.8c for vacancy islands of 100 to 800 spacings, correlated or uncorrelated evaporation–condensation was suggested, while for bigger islands the limiting step was believed to be terrace diffusion. Monte Carlo simulations for diffusion of copper islands on Cu(100) were carried out by Heinonen et al. [13] drawing on the EMT energetics of Merikoski and Ala-Nissila [14]. For the diffusion as a function of the number of cluster atoms N they obtained the plot in Fig. 9.9a at a temperature of 1000 K. Islands with more than ten atoms gave a straight line, which yielded the exponent plotted in the inset. This varied from the value for edge diffusion around small clusters to that for terrace diffusion limiting when the cluster became large. That is, the mechanism of diffusion changed with size, but the

671

9.1 Large clusters on fcc(100) surfaces

Table 9.1 Activation energies for diffusion of silver clusters calculated for Ag(100) plane [15]. Process

Activation energy (eV)

Process

Activation energy (eV)

KD TC EA PD

0.55 0.60 0.75 0.25

M VM TCPD TPD

0.45 0.55 0.50 0.35

1.25

108 1 100

104 102 Cluster size, N

106

D(a2 /sec)

αN

D(a2 /sec)

1010

1.5

1010

105

100

(a) 104 100

Fig. 9.9

(b) 101

102 Cluster size, N

103

104

10–5 100

101 102 Cluster size, N

103

104

Cluster diffusivity on Cu(100) derived from EMT energetics. (a) As a function of size N at T = 1000 K. (b) Diffusivity as function of size at temperatures 1000 K, 700 K, 500 K, 400 K, 300 K, starting at the top (after Heinonen et al. [13]).

coefficient did not strongly depend upon the temperature. For this system the activation energy for diffusion did not change much on going from edge diffusion to terrace diffusion being limiting. They also observed a crossover from periphery dominated transport to vacancy dominated diffusion inside the clusters. At low temperatures, the diffusivity oscillated with increasing size, as shown in Fig. 9.9b. Movement of vacancy islands was found to be very similar to that of adatom islands. The same year Mills et al. [15] employed Monte Carlo simulations to look at islands on an fcc(100) plane, with parameters to mimic Ag on Ag(100); energetics were obtained from DFT calculation, or from EMT scaled to DFT values. Barriers not available from DFT or EMT were estimated from bond counting, with the strength of one bond at ~0.3 eV. The energetics in the simulations are listed in Table 9.1. It is quite surprising that the energy required for adatom motion along a straight step, 0.25 eV, is much smaller than the energy for isolated adatom movement over the (100) surface, 0.45 eV. This suggests that mass transport across the terrace was less favorable than transport along the edges, so that the evaporation–condensation mechanism would be unlikely to happen. Fig. 9.10a shows the processes considered in this study. The power law dependence of diffusivity D on island size N was described by D  expð
Ep =kTÞN
αN

(9:3)

where Ep denotes the activation energy for movement along steps. They found that the power law was valid only for large clusters, which satisfy the relation Nl < N < 2000,

672

Diffusion of large clusters

103 0.59 eV

TPD KD

PD

102

M

101 TC

1

VM

TC TCPD

D (1010cm2/sec)

10–1 10–2

(a)

EA

2000 atoms

10–3 10–4

0.79 eV

10–5 10–6

TPD

250 atoms

0.57 eV

10–7 0.5

(b)

0.69 eV

1

1.5

2

2.5

3

3.5

1000/T (K–1)

Fig. 9.10

Diffusion of clusters on fcc(100) surface from MC simulations. (a) Illustration of possible atomic steps in diffusion. (b) Arrhenius plots for clusters of 250 and of 2000 atoms. Prefactors increase at low temperatures (after Mills et al. [15]).

where Nl depends on temperature; it is 200 at temperatures above 450 K, and 300 below 450 K. They did simulations at 1000 K with an island of 125 atoms and also 2000 atoms with 20 to 30 atoms in the vapor phase under two boundary conditions: first with evaporation–condensation forbidden, second with peripheral motion forbidden. During simulations with evaporation–condensation not allowed, the diffusion coefficient reached 90% of its value when all events were active, showing that evaporation– condensation influences the diffusivity very weakly. At the same time, when peripheral motion was forbidden, the diffusivity reached only 10% of the overall value. This study clearly indicated that diffusion was driven mostly by adatoms moving along the edges. Mills et al. [15] also looked at the temperature dependence of the diffusivity for islands of 125 and 2000 atoms. Their findings are presented in Fig. 9.10b. We clearly see the situation changing from low to high temperatures. Of course the measured diffusivity is an average result over all processes participating. However, two distinct regions in the diffusivity allowed them to speculate that the mechanism will be caused mostly by two main processes, of which one will dominate at low temperatures and a second one at high temperatures. The task would be to decide which processes are involved. At high temperatures, the energy obtained from the Arrhenius plot was 0.57 eV, and a prefactor of 1.7 × 10–7cm2/sec for the 2000 atom island, and 0.59 eV, with a prefactor of 5.5 × 10–6 cm2/sec for the 250 atom island. Both values are comparable to the energy of corner or kink breaking, also called core breaking, which might be the limiting step in the motion. The situation is less clear at low temperatures, where the energy from the Arrhenius plot was much too high for edge hopping or core breaking. The values are 0.69 eV, and the prefactor amounted to 3.6 × 10–6cm2/sec for the 2000 atom island and 0.79 eV with a prefactor of 6.3 × 10–4cm2/sec for the 250 atom island. A possible rationalization could be nucleation of new rows with an effective energy of 0.81 eV, but it is not clear what really is responsible for such a high energy.

9.1 Large clusters on fcc(100) surfaces

673

That cluster diffusion occurs by atom movement around the periphery was further confirmed in a subsequent study by Pai et al. [16] of worm-like vacancy clusters. These were created during deposition of an incomplete layer (~ 0.6 ML) of copper on Cu(100) and silver on Ag(100), after allowing the clusters to collapse to create percolation vacancy islands. Since worm-like vacancy islands are quite irregular, their reshaping allows us to get insight into the diffusion mechanism. The frequent pinch-off observed was found to be distinctive for atom diffusion around the cluster edges, and speaks against any of the other mechanisms. Comparison of the experimentally evaluated shape of the vacancy islands with simulations confirmed that reshaping is a geometry driven process. The same year Resende and Costa [17] examined the deposition of Cu13 clusters on the Cu(100) surface using molecular dynamics and Monte Carlo simulations, relying on interactions based on a second-moment approximation to tight-binding. Following the movement of clusters over the surface they found that clusters maintained their integrity even at 800 K during diffusion and moved by single hops of the peripheral atoms. Specific attention was paid by Wang et al. [18] to so-called “diagonal atom motion” in the diffusion of clusters, again using Monte Carlo simulation with EMT energetics. Diagonal motion is just the movement of an atom around a cluster corner, as illustrated in Fig. 9.11a1 together with other hopping processes taken into account in these studies. The diagonal movements are marked in Fig. 9.11a as TPD and TCPD. They occur over an additional barrier ET that has to be overcome for such movements, compared to ED for free atom hopping over the substrate. The effect of such transitions on the size dependence of the diffusivity, D / N
αN , is shown in Fig. 9.11b at 300 K for clusters of 15–300 atoms. The deviation observed for small clusters can be caused by fluctuations in the cluster size or changes in the shape; that is why Wang et al. took into account only data for N > 40. The exponent αN changes as the relative magnitude of the diagonal barrier ET changes. As indicated in the inset, αN ¼ 1.54 ± 0.034 if ET = 0, and it falls to 1.03 ± 0.027 eV when ET = 1.55 ED. How the different diffusion mechanisms are affected by changes in the magnitude of the barrier to diagonal motion is indicated in Fig. 9.11c, where T describes diagonal motion, PD perimeter and kink movement, EC evaporation– condensation, and CEC the correlated evaporation–condensation mechanism. The biggest contribution when diagonal motion requires too much energy is made by peripheral diffusion in combination with detachment from kinks, as suggested in the schematic in Fig. 9.11d. From this study it is clear that as long as the energy of diagonal motion is low, ET < 0.4 ED, motion around the corners is realized by diagonal movements T and αN → 1.5; however, when the energy for diagonal movement increases, ET > 0.4 ED, then corner crossing is done by the correlated evaporation–condensation mechanism CEC and αN → 1.0. Jahma et al. [19] used Monte Carlo simulations for copper island diffusion on both Cu (100) and Cu(111) surfaces, based also on energetics from effective medium theory. Results for the diffusivity on Cu(100) as a function of size, given by the number of atoms N, and the temperature are shown in Fig. 9.12b. Here, as in previous studies, there were 1

See also Fig. 9.7a.

674

Diffusion of large clusters

EA

10–16

CD

1.4

αN

TCPD

10–17

1.2

CPD

D (cm2/sec)

1.0

KA

10–18

0.0

1.2

1.6

10–19

ED ED

10–21

KD

ED

ED

10

TPD

100 1000 Cluster Size, N

(b)

PD

102 Percentage (%)

0.8

ET /ED

10–20

(a)

0.4

PD + KD(KA)

101 EC

KD

100

PD

KA effective CEC

10–1

(d)

10–2 T

(c)

10–3 0.0

0.4

0.8

1.2

ED

Cluster diffusion on fcc(100) surface derived from MC simulations. (a) Elementary diffusion events on cluster periphery. (b) Dependence of cluster diffusivity at 300 K on cluster size N. Inset shows dependence of scaling coefficient αN on ED. (c) Occurrence of different mechanisms as a function of the change in the diagonal motion energy, T – diagonal movement such as TPD and TCPD. (d) Model for rate-limiting step in cluster diffusion (after Wang et al. [18]).

Diffusivity, D (a 2 /sec)

1012

1012

300 K 500 K 700 K

109

Diffusivity, D (a2 /sec)

Fig. 9.11

106 103 100 (a)

10–3 100

Fig. 9.12

109 106 103 100

10–3 101 Cluster size, N (atoms)

102

300 K 500 K 700 K

100

(b) 101 Cluster size, N (atoms)

102

Diffusivity of adatom islands in its dependence upon size N and upon temperature based on EMT. (a) On Cu(111). (b) On Cu(100) (after Jahma et al. [19]).

675

9.1 Large clusters on fcc(100) surfaces

0.0005 re = 1

Diffusivity D

0.0004

rc = 0.2

0.0003

0.0002

rc = 0.1

0.0001

rc = 0.05

0 0

Fig. 9.13

0.002 0.004 0.006 0.008 1/N

0.01

Diffusivities of Ag clusters on Ag(100) surfaces obtained in MC simulations in their dependence upon N−1, where N gives the number of cluster atoms. rc is the rate of evaporation–condensation, re that of peripheral diffusion (after Sa´nchez [20]).

significant oscillations with size, especially at lower temperatures. At the higher temperature of 700 K, these oscillations became much less significant and the diffusivity was reasonably represented by the curve D / N
αN , with αN = 1.5, the value if the limiting step was atom diffusion around the island edges. The activation energy for atom diffusion was 0.4 eV on Cu(100). Monte Carlo simulations of cluster diffusivities were done for (100) surfaces by Sa´nchez [20], who examined mobilities as a function of rc/re, the ratio of the rate of evaporation–condensation to the rate of diffusion around the periphery. One interesting result is shown in Fig. 9.13, where the diffusivity is plotted against N−1, N being the number of atoms making up the cluster. For different values of the evaporation– condensation rate, the cluster diffusivity is a linear function of N−1, and has a greater value as rc/re increases. Sa´nchez concluded that although the predominant process was peripheral diffusion, evaporation–condensation had a pronounced effect on the diffusivity. However, experimental measurements have been obtained mostly around room temperature, at which evaporation could not occur, leaving peripheral diffusion as the dominant mechanism [8,10,16,21]. Studies of large clusters on fcc(100) surfaces predict movement by edge running. The evaporation–condensation mechanism is unlikely, but possible in some specific situations, for example where corner movement is blocked by a huge barrier [18]. Most of the MC simulations relied on the same EMT energetics, which might also lead to misleading results. It is very difficult to judge how all this applies to real materials, since measurements have been mainly done on the Ag(100) surface, and even here deriving the mechanism has not been an easy task. Different materials might require different starting parameters. What is surprising is that there are no investigations concerning the influence of dimer or trimer shearing on the movement of large clusters, mechanisms which are clearly present in the movement of smaller clusters on fcc(100). For fcc(100), there is also

676

Diffusion of large clusters

no indication of the existence of stable forms that move by gliding and which are seen on fcc(111) surfaces; stable clusters are probably a characteristic feature of smooth surfaces.

9.2

Large clusters on fcc(111) surfaces

1000

Fig. 9.14

(a) 800 600 400 200 0

0

50

100

150 200 250 Time (sec)

300

350

Mean-square displacement (Å2) for time = 60 sec

Mean-square displacement (Å2)

Diffusion of pits and large clusters on Au(111) surfaces in an electrochemical environment was discussed by Trevor and Chidsey [22] in 1991. They used scanning tunneling microscopy and found that “pits, islands and step edges” diffused at room temperature. They proposed a model in which atoms diffuse around the pit edge, moving the center of mass. For this scheme the diffusivity was proportional to Rr−3, where Rr gives the radius of the pit. Indeed, small pits, 20–40 Å in diameter, moved very rapidly, whereas large pits 100 Å in diameter were almost stationary. Based on these observations, they estimated an upper bound for the free energy to create an adatom from the step, together with an energy for adatom diffusion of 0.5 eV. Vacancy islands on Cu(111) covered with a partial monolayer of cobalt were examined a few years later by de la Figuera et al. [23] again using a scanning tunneling microscope under UHV conditions. Vacancy islands were created in a controllable way relying on tip–surface interactions. Islands ranged in size from just a few vacancies to voids with a diameter of 75 Å. In diffusion, the islands maintained their hexagonal shape. The meansquare displacement of individual vacancy islands proved to be linear in time as shown in Fig. 9.14a, in agreement with the notion of Brownian displacements. From measurements on differently sized islands they were able to construct a plot of the mean-square displacement against the island area, shown in Fig. 9.14b. As random motion, the meansquare displacement was approximately inversely proportional to the area of the islands. On a clean Cu(111) surface, however, Cu vacancy islands were essentially not mobile at room temperature, unless strain was present . Morgenstern et al. [24] using the STM looked at vacancy islands on Ag(111). Diffusion of these clusters is shown in Fig. 9.15. They derived an expression for the dependence of the diffusivity for atoms running around the perimeter and for

(b)

1600 1200 800 400 0

0

500

1000 1500 2000 2500 3000 Area (Å2)

Diffusion of vacancy islands on Cu(111), covered with a partial layer of Co, observed with STM. (a) Mean-square displacement as a function of time. (b) Mean-square displacement at 60 sec in its dependence on area of vacancy islands (after de la Figuera et al. [23]).

677

9.2 Large clusters on fcc(111) surfaces

(a)

(b) 0.6

D (nm2/sec)

0.4

0.2

D (nm2/sec)

1

(c)

αL = –1.97±0.39

0.1 0.01 10

LL (atoms)

100

D~1/LL2 0.0 10 20 30 40 50 60 70 80 90 100 LL (atoms)

Fig. 9.15

Diffusion of vacancy islands on Ag(111). (a)–(b) Motion as observed with STM. (c) Diffusivity of vacancy clusters as a function of cluster diameter (after Morgenstern et al. [24]).

evaporation–condensation. For cluster motion by atom diffusion around the edges they arrived at a diffusivity proportional to the inverse of the cluster diameter raised to the power three. For evaporation–condensation, this dependence changed to LL−2, where LL is the diameter. Measurements were done under UHV conditions, and the mean-square displacement was again found to vary linearly with time, as expected for a random process. Mobility of vacancy islands was already high at room temperature, with bigger islands moving slower. The dependence of the diffusivity D on the size of the vacancy cluster is shown in Fig. 9.15c. As indicated in the inset, the diffusivity is given by D / LαLL , where αL ¼
1:97  0:39, in agreement with atom diffusion over the terrace rather than along the edges. What is interesting is that Morgenstern et al. never observed island coalescence, even if the islands came as close as five atomic distances. These interesting findings have been examined again briefly, in view of a final summary by Rosenfeld et al. [25] about the diffusivity of vacancy clusters on Ag(111). In the above studies, they had found a diffusivity given by D / LL−1.97 ± 0.39, which is expected for evaporation–condensation. However, they subsequently measured the rate of evaporation of atoms from cluster edges and found this was on the order of 1 sec−1; at the same time, they estimated from the prefactor of the diffusivity a rate three orders of magnitude higher, amounting to 750 sec−1. Furthermore, in Monte Carlo simulations of the movement of vacancy clusters on Ag(111), with sizes and energetics comparable to the experiments, they were able to agree with the experimentally found scaling coefficient of two. However, they also discovered that their results were not significantly

678

Diffusion of large clusters

affected when evaporation was forbidden, so that only diffusion around the perimeter could occur. They concluded that diffusion around the edges of the vacancy cluster was really the mechanism for motion, even though the diffusivity did not vary as D / 1/LL3, as expected for peripheral atom motion, and that further studies were in order. In 1995 Hamilton et al. [26] introduced a new mechanism for cluster diffusion on fcc(111) planes, already described in Section 7.2. Part of the cluster, in their scheme, moved from fcc to hcp positions, with the two parts separated by a misfit dislocation. This could move across the cluster, resulting in diffusion. This mechanism was shown to be favorable for clusters of 20 to 100 atoms. For movement of clusters smaller than 20 atoms, gliding was predicted as the leading mechanism, while for clusters bigger than 100, movement by kink motion was expected. They showed that the dislocation mechanism was more probable than adatom edge running. No attempt to calculate diffusivities with this new mechanism was made. A year later, Hamilton [27] did further investigations testing his mechanism for the hetero-motion of silver clusters on Ru(0001) surfaces. Ruthenium is an hcp metal, but the (0001) surface looks very similar to fcc(111). Using EAM, he calculated activation energies at 0 K for diffusion of 19, 37, 61, 91, and 127 atom clusters by dislocation movement; the energies were respectively 0.681 eV, 0.729 eV, 0.457 eV, 0.293 eV, and finally 0.786 eV. From a molecular dynamics study he found that diffusion of a 61 atom cluster around room temperature occurred at higher rates than 37 and 91 atom clusters. This happened due to a change of the activation energy for movement with temperature, arising from the temperature dependence of the misfit between the Ag cluster and the Ru(0001) surface, as shown in Fig. 7.28. The minimum activation energy for 61 atom clusters was as low as 0.18 eV at ~309 K. The diffusion of xenon clusters, a non-metal, on Pt(111) was examined by Monte Carlo simulation in the hands of Sholl and Skodje [28], who showed that evaporation– condensation was the preferred mechanism by which vacancy as well as adatom clusters diffused. Bitar et al. [29] used Monte Carlo simulations to look at islands 10 to 100 atoms big, and resembling gold on an fcc(111) surface. The diffusivity decreased with the size of the island according to a power law, with an exponent equal to −2.1 ± 0.1 for clusters with N > 20; for clusters smaller than 20 atoms the exponent was −1.8 ± 0.1. The authors associated the exponent with the fast movement of atoms along the edges. They also indicated that changes in the exponent between 1.5 and 2 can be observed due to the choice of interactions, including taking into account next-nearest-neighbor effects. From the temperature dependence of the diffusivity the activation energy for movement, ED D, was derived, and turned out to be size independent and close to the experimentally derived value for removing an atom from a kink of the Au(110) surface [30]. Values Bitar et al. obtained from Arrhenius plots of the diffusivity, as well as from the cluster velocity, EvD , are presented in Table 9.2. The diffusion of large clusters on surfaces with an fcc(111) structure without misfit was reexamined by Bogicevic et al. [31]; they did Monte Carlo simulations allowing only diffusion of atoms around the cluster edges. In these simulations two different energetics were considered, illustrated in Fig. 9.16a; all rates had a pre-exponential factor of 1013 sec−1. The results for the diffusivity at different temperatures as a function of the

679

9.2 Large clusters on fcc(111) surfaces

Table 9.2 Energetics of clusters from MC simulations on fcc(111) surfaces [29]. Size of island N (atoms)

ED D ðeVÞ

EvD ðeVÞ

10 20 30 40 100

0.59 ± 0.07 0.59 ± 0.06 0.61 ± 0.06 0.59 ± 0.04 0.60 ± 0.03

0.60 ± 0.09 0.61 ± 0.09 0.62 ± 0.08 0.64 ± 0.07 0.63 ± 0.05

5 6 7

0.4 0.85

0.6

0.45

4 1

2

0.7

6

3 5

Fig. 9.16

0.45

7

Diffusivity, D(10–10cm2/sec)

1 2 3 4

(a)

1000 (b) 100

T = 700 K, Model1 T = 700 K, Model2 T = 1300 K, Model1 T = 1300 K, Model2

10 1 0.1 0.01 10

1000 100 Number of atoms in cluster, N

Edge diffusion of adatoms around cluster on fcc(111) surface obtained in MC simulations. (a) Top: model of edge structure. Bottom: Activation energy (in eV) for different atomic movements. Second model arises from different activation energies. (b) Dependence of island diffusivity at 1300 K on the number of atoms in the cluster (after Bogicevic at al [31]).

number of cluster atoms are shown in Fig. 9.16b, and it is clear that when the number of cluster atoms exceeds 100 there is an exponential relation with size. However, the exponent is not a constant, as suggested previously. Instead, it varies with the temperature and with the energetics of atom diffusion around the edges, which indicates a material dependence. For clusters much larger, with 1000 to 2000 atoms, the exponent was almost constant and differed by less than 1%. The same year, Bogicevic et al. [32] using density-functional calculations and kinetic Monte Carlo simulations, mapped out the energetics for movement of large clusters on an Al(111) surface. The processes considered are presented in Fig. 9.17a and the energies derived are given in Table 9.3. What is interesting to see is that movement along B-type steps proceeded by exchange rather than by hopping; exchange was also the preferred mechanism for kink incorporation. On an A-type step hopping is the leading mechanism, but exchange was in operation over this range of temperatures as well. Bogicevic et al. also observed a huge anisotropy for corner diffusion; the corner energy barrier to B-type steps was 400% higher than to A-type steps. Based on the energies derived from DFT they constructed a scale showing the increasing number of processes activated with temperature, as given in Fig. 9.17b. From 0 to 120 K, only terrace and corner diffusion is active, so an atom randomly deposited next to a corner will relax to a position next to the step. Islands are not compact, as movement along edges was suppressed, but a gradual

680

Diffusion of large clusters

Table 9.3 Energetics of self-diffusion processes from LDA/GGA in Fig. 9.17 as a function of aluminum cluster size N [32]. Process

N

EA h

EA e

EBh

EBe

E2 → 2 K3 → 2 K2 → 3 C1 → 1 C1 → 2 C2 → 1 C3 → 1 C1 → 3

109–190 135–192 135–192 129 129 129 131 131

0.31/0.31 0.47/0.45 0.27/0.28 0.18/0.18 0.04/0.05 0.33/0.33 0.62/0.59 0.03/0.04

0.43/0.36 0.46/0.38 0.26/0.22

0.45/0.45 0.67/0.65 0.41/0.42 0.22/0.22 0.17/0.19 0.32/0.30 0.63/0.60 0.14/0.14

0.35/0.26 0.49/0.42 0.24/0.19

h-hop, e-exchange

E2–>0

Edge evaporation 300

C3–>1

K3–>2

C2–>1 E2–>2

T0–>0

C3–>1

250

Corner breaking

A-step C1–>3

200 K3–>2

K2–>3

Kink 150 breaking

B-step C1–>1

C1–>2

E2–>0

C2–>1

E2–>2

Edge diffusion C1–>1

K2–>3

100

Corner crossing CB1–>2

(a)

C1–>3

Kink incorporation

Corner flipping 50

Corner incorporation

C

T0–>0

Fig. 9.17

1–>2

Corner diffusion A

Terrace diffusion 0

(b)

Corner diffusion B

A

°K

Self-diffusion of clusters on Al(111) derived from density-functional calculations. (a) Schematic of participating diffusion processes. (b) Temperature scale for onset of diffusion events (after Bogicevic et al. [32]).

681

9.2 Large clusters on fcc(111) surfaces

Ir19 675 K

Ir18 550 K

(a)

(b)

Ir33 650 K

(f) Fig. 9.18

Ir24 600 K

(c)

Ir36 650 K

(g)

Ir26 600 K

(d)

Ir37 700 K

(h)

Ir27 650 K

(e)

Ir38 675 K

(i)

Ir39 675 K

(j)

Iridium clusters observed on Ir(111) plane in FIM together with annealing temperature [33].

transition towards less of a fractal shape proceeds. At 120–170 K, a transition begins to compact clusters, arising from edge movement and corner crossing. At 170–325 K the shape changes to islands with an equilibrium form. At higher temperatures, edge evaporation and exchange of atoms between neighboring islands starts playing a role and Bogicevic et al. [32] claimed good agreement with unpublished experimental observations by Fischer et al. in Kern’s laboratory in Lausanne. Questions concerning cluster diffusion were tackled at roughly the same time by field ion microscopy, which can directly reveal the individual atomic events. Studied by Wang and Ehrlich [21,33] were clusters of iridium on Ir(111), shown in Fig. 9.18 after equilibration, with sides of unequal as well as equal length. In none of these did evaporation intervene in diffusion. With Ir18, Ir26 as well as Ir36, diffusion occurred at T ≥ 550 K by atoms moving along the cluster edges, which are seen in Fig. 9.19 to be of unequal length. The diffusive movement proceeded without any additional adatoms on the terrace and with the number of cluster atoms conserved, ruling out the possibility of evaporation–condensation. Changes in the shape of these clusters were directly observed in the FIM; the individual atom jumps, however, could not be detected. It was always two

682

Diffusion of large clusters

I

II

III

IV

Fig. 9.19

Ir18 clusters of non-equivalent form on Ir(111) after equilibration at ~ 550 K observed in FIM [21].

Fig. 9.20

Direct FIM images of Ir clusters diffusing on Ir(111) [33]. (a)–(d) Compact Ir19 diffusing at ~675 K without shape change. (e)–(h) Ir38 cluster diffusing at 400 K with position of attached Ir atom constantly altering.

atoms with Ir18 and three with Ir36 clusters that displaced. At a temperature of ~700 K, evaporation of atoms from non-compact clusters was observed; the free atom never came back to the cluster, rather it diffused over the terrace. Close-packed hexagonal Ir19 clusters also moved over the surface, shown in Fig. 9.20a–d. This diffusion differed from Ir18, however, in that motion of the hexagonal cluster occurred at temperatures more than 100 K higher, without any apparent movement of atoms around the cluster periphery, and without shape changes. What also became clear from observations of the cluster displacement was the participation of long jumps, that is transitions larger than a displacement between nearest neighbors. The movement of Ir37, another compact cluster, also proceeded without any shape changes; it must be noted that this cluster was more stable than non-compact ones. An Ir38 cluster created by adding an atom to Ir37 is shown in Fig. 9.20e–h. The movement of the additional atom around the stationary, hexagonal core consisting of 37 atoms proceeded on heating to 400 K. The atom was always observed next to type A steps, indicating stronger binding there compared with B-type steps. The Ir38 cluster started changing its shape when the temperature was increased to 700 K, as in Fig. 9.21. Analyzing the frequency at which different cluster shapes appeared, it was possible to draw the

683

9.2 Large clusters on fcc(111) surfaces

Fig. 9.21

Views of Ir38 cluster on Ir(111) in FIM after annealing at ~ 700 K [33].

conclusion that for other than compact clusters, shapes with longer B-type edges are preferred over those with A-type edges. That makes binding at A-type edges stronger than at steps of type B, which agrees with observations of the presence of an atom attached only to step A of Ir38 at temperature 400 K, shown in Fig. 9.21e–f. From the frequency of observation for different cluster shapes it is possible to derive the free energy change for converting one shape into another, according to DF ¼
kT ln½ðNI =OI Þ=ðNII =OII Þ;

(9:4)

where NI and NII denote the frequency of occurrence of forms I and II, while ΩI and ΩII denote the degeneracy, the number of equivalent configurations of the two cluster forms. A rough estimate of the free energy per atom at a specific type of edge showed that at a step of type A the average free energy was 0.029 ± 0.008 eV higher than for an adatom at a straight step of type B. From the frequency of observation of kinked clusters it was deduced that this structure had the lowest energy for Ir26 and Ir39. For Ir18 and Ir36, the difference between the free energy of kinked and more stable forms was very small. For hexagonal clusters, the free energy for other shapes exceeded 0.025 eV. From a study on Ir(111) surfaces it was possible to divide large clusters into two groups: a hexagonal one, for which the number of atoms Nc can be written Nc ¼ 1 þ 3sðs
1Þ;

s ¼ 2; 3 . . .

;

(9:5)

and all the others. Clusters with the number of atoms given above are more stable and remain in a hexagonal shape, while others change their shape and are less strongly bound. For example, Ir19 remains hexagonal until it disappears at 700 K, while Ir18 frequently changes shape and on heating to 600 K dissociates. Clusters such as Ir39 after prolonged heating at 700 K dissociate, leaving a stable Ir37 on the surface. More quantitative studies were subsequently done by Wang et al. [34], who measured the mean-square displacement at different temperatures. Shown in Fig. 9.22a,b are plots for diffusion of Ir18 as well as Ir19 on the Ir(111) plane. What is notable here is that for Ir18 clusters with edges of unequal length, the prefactor for the diffusivity is much the same as for a single iridium adatom; the activation energy for diffusion, 1.63 ± 0.07 eV for Ir18, is of course much higher than for individual Ir adatoms. Diffusion was also examined for two hexagonal clusters of different size, Ir19 and Ir7. It is clear that both have unusually high prefactors, 13(× 2.6±1) cm2/sec for Ir19 and

684

Diffusion of large clusters

Fig. 9.22

Temperature dependence of the diffusivity for Ir clusters on Ir(111) plane observed in FIM [34]. (a) Arrhenius plot for diffusivity of Ir18 cluster. (b) Arrhenius plot for diffusivity of Ir19. (c) Distribution of center-of-mass displacements for Ir19 cluster during 10 sec at 690 K. Bold letters give observations, best fit in outline numerals below, to the left. Fit with nearest-neighbor jumps only is at the bottom right.

1.4(× 2.4±1) cm2/sec for Ir7; the diffusion barrier amounted to 2.54 eV for the larger cluster and 1.49 eV for Ir7. The distribution of cluster displacements for Ir19 was also measured and is shown in Fig. 9.22c, in which are indicated the three types of jumps identified in diffusion. What is unusual is that the long jumps, double β’s and triple γ’s occurred at a considerable rate and contributed ~60 % to the diffusivity at 690 K. For Ir7,

685

9.2 Large clusters on fcc(111) surfaces

106 105 0.35

10

104

Krylov prefactor = 6000 barrier = 7.0

103

0.15

2

10

10

(a)

0.1

0.001

(b) 0

Vineyard prefactor = 6.1 barrier = 5.5

0.01

0.0001 0.9

1

Fig. 9.23

Hopping Rate

P (N )

1

Plotted Points Molecular Dynamics

1

1.1 1.2 1.3 1.4 1.5 1/kT

1.6

1.7

2 4 6 8 10 12 14 16 18 20 Number of atoms is cluster, N

Kinetics of cluster diffusion. (a) Predicted dependence of diffusivity prefactor on the number N of atoms in a cluster for frequency ratios νðcÞ =ν0 = 0.15 and 0.35 (after Krylov [35]). (b) One-dimensional diffusion of Ir chains. Top: stretched and compressed transition states in one-dimensional diffusion. Bottom: Arrhenius plots for hopping rate of Ir seven atom chain over a periodic potential (after Hamilton and Voter [36]).

long jumps are not of comparable importance. Clearly diffusion of these hexagonal clusters occurs in an unusual manner, and it must be emphasized that the big jump lengths are not responsible for the large values of the prefactors. The mechanism of movement was not revealed. However, it is possible to disallow the evaporation–condensation mechanism, since an Ir18 cluster is unstable at the temperature at which compact clusters move, so detachment of one atom will result in dissociation of the entire cluster. In an attempt to describe the surface diffusion of hexagonal, close-packed clusters, which was found by Wang et al. [34] to occur with a very large prefactor and may also have involved long jumps, Krylov [35] came up with a one-dimensional theory for cluster gliding. He was seeking a rationalization for the huge prefactor caused by internal vibrations and arrived at a diffusivity D given by D ¼ Do PðNÞ expð
ED =kTÞ;

(9:6)

where the additional term P(N) in the prefactor arises from a dynamical misfit. This term is close to unity when the number of atoms N in the cluster is small, but rises exponentially as N increases. A plot of this dependence on the number of atoms is given in Fig. 9.23a, for two values of the vibrational frequency ratio νðcÞ =ν0 upon which P(N) depends, and it is clear that clusters with tens of atoms may indeed have a prefactor much larger than normal. Krylov also advanced a separate kinetic argument for a rise in the number of long transitions in diffusion and associated this effect with size dependent cluster–substrate coupling, caused by interactions of short phonon wave length with intra-cluster vibrations. This theory has been strongly criticized by Hamilton and Voter [36], who worked out the rate of seven atoms chain hopping by molecular dynamics for one-dimensional motion as well as by resorting to Vineyard’s [37] expression for the rate of jumping. Both give a prefactor on the order of unity, as shown in Fig. 9.23b. Krylov [38], however,

686

Diffusion of large clusters

Table 9.4 Prefactors for cluster diffusion calculated with GGA potentials on Ir(111) in units of sec−1 [40]. Mechanism

1 atom

7 atoms

19 atoms

Hopping 2 × 1012/3 × 1012 Bridge glide 7 × 1013/1 × 1014 2 × 1016/7 × 1016 Cartwheel shuffle 7 × 1014/5 × 1014 8 × 1016/1 × 1017 11 Experiment 7 × 10 3 × 1015 2 × 1016

Reference [41]/ [42] [41]/ [42] [41]/ [42] [34]

rejected these arguments, claiming that molecular dynamics simulations fail due to the use of only one spring constant, which might not represent the physics correctly. In the Vineyard approach, the minimum energy barrier was used instead of an averaged effective one for all transitions. This leaves unsettled an explanation for the magnitude of the cluster prefactors. That question has been taken up in theoretical studies; we first mention work by Kürpick et al. [39], who examined Ir7 clusters on Ir(111) more closely with Sutton–Chen potentials. Three sequences from simulations at 1350 K are shown in Fig. 7.24. The clusters move by slightly different paths from one energy minimum to a neighboring position. For the concerted displacement from fcc to hcp sites, a potential energy increase of 1.23 eV was found with Sutton–Chen potentials and 1.51 eV using RGL; several non-equivalent mechanisms for cluster diffusion were identified with activation energies comparable to the experimental value of 1.49 ± 0.03 eV. Prefactors for the different mechanisms were evaluated and turned out to be in the usual range of 10−4 cm2/sec. Her conclusion was that there exist a number of non-equivalent paths for diffusion, with activation energies comparable to the measured value. Prefactors of the magnitude usual in single adatom diffusion were found in simulations; there was no indication of any unusually large value for any single path. However, the number of mechanisms was large, and they allowed the cluster to move without any significant change in shape. Their superposition created the large overall prefactor found in the experiments. The intriguing mode of diffusion for compact hexagonal Ir7 and Ir19, with very high prefactors, high activation energies, and long jumps, was addressed again by Hamilton et al. [40], using generalized gradient approximations and the nudged elastic band method. They found that a pseudomorphic monolayer had a fairly low energy metastable minimum when in the on-top location on a crystal. Semi-empirical methods, such as EAM, EMT or Sutton–Chen, did not detect this minimum. With this result as a motivation, they considered four different mechanisms of cluster diffusion shown in Fig. 7.29: glide over bridge sites, glide over on-top positions, as well as two combinations of cluster rotation and translation: cartwheel shuffle and cartwheel glide. The results of their calculations for both Ir7 and Ir19 are given in Table 9.4, but it should be noted that for Ir19, calculations were limited. For Ir7, bridge gliding is in very good agreement with the measured barrier of 1.49 ± 0.03 eV [34]. The situation with Ir19 clusters is different, as now bridge glide and cartwheel shuffle have much higher calculated barriers, 3.2 eV and 3.6 eV respectively, compared to the 2.54 ± 0.06 eV found in experiments, but this could have stemmed from calculational approximations. Hamilton et al. believed that

687

Energy (eV)

9.2 Large clusters on fcc(111) surfaces

Fig. 9.24

4 3 2 1 0

fcc

metastable

hcp

Energy calculated using generalized gradient approximation for translation and rotation in cartwheel shuffle of Ir19 cluster on Ir(111) plane (after Hamilton et al [40]).

cartwheel shuffle (possibly combined with bridge glide) should be considered in attempting to account for the long jumps. The energy change during a cartwheel shuffle is shown in Fig. 9.24, and it is clear that there is a very broad metastable position, but that the energy changes rapidly at both fcc and hcp sites. In the Vineyard [37] formulation of rates the prefactor is given by the product of vibrational frequencies in the fcc state over the product in the transition state. Judging from the potential diagram, the latter vibrations should have much lower frequencies, which should give a higher prefactor. To estimate this, calculations were done with Sutton–Chen [41] and Chen [42] potentials, giving the results in Table 9.4. Higher prefactors than usual were in fact obtained. Monte Carlo simulations of the diffusion of Ir18 clusters on Ir(111) have also been made by Zhuang and Wang [43] relying on the energetics of Bogicevic et al. [31]. Presumably diffusion occurred via the peripheral movement of individual atoms around the edges, as is suggested in the diagram in Fig. 9.25a. In field ion microscopic observations, however, two atoms were always seen to transfer from one position at the edge to another [34], probably because single atom events took place too rapidly, making it impossible to observe the intermediate shapes shown in Fig. 9.25a as states a–e. Zhuang and Wang therefore decided to see if the experimentally observed results could be simulated by jumps of an effective dimer, as is suggested in Fig. 9.25b, where the barrier heights were obtained by fitting to the experiments. As shown by the temperature dependence of the diffusivity, plotted in Fig. 9.25c, agreement was very good, as the experimentally derived activation energy was 1.63 ± 0.07 eV with a prefactor 7.8 (× 4.5±1) × 10−4 cm2/sec. This study shows that dimer shearing movement is most likely responsible for movement of Ir18 cluster. In 2000, Chirita et al. [44] extended their considerations of cluster diffusion on Pt(111) by sub-unit shearing to higher clusters. In their work they relied on the embedded atom potentials of Johnson and Oh [45,46], and worked on platinum clusters of 6 to 25 atoms. An example of their proposed mechanism is shown in Fig. 9.26 for Pt11. Starting with all atoms in fcc sites, 5 atoms glide to hcp sites, which creates a stacking fault. The motion is completed by relaxation of the remainder into hcp positions. For hexamers, they estimate a barrier to diffusion of ’1.55 eV, and contend that double (reptation) shearing should “replace concerted cluster gliding as the energetically favored migration mechanism with increasing cluster size.” For Pt6, Pt7, and Pt19 activation energies of 1.22 eV, 1.50 eV, and 4.37 eV were estimated for cluster diffusion, in which all atoms make concerted jumps. If the number of cluster atoms is less than 7, concerted glide and double shearing are competitive, but for larger clusters the latter is favored according to their study. Shearing

688

Diffusion of large clusters

3.0

(a)

I

2.0

a

34.79 0

1.5

b

c

1.21

2.16

d

e

III

1.22

0

60.61

1.0 0.5 0.0

I

I

III

III

II

III

IV

Diffusivity D(10–16cm2/sec)

Energy (eV)

2.5

0.1

Do = 7.4(×3.1±1) ×10–4cm2/sec

0.01

(c)

IV 0.001

(b)

ED = 1.62±0.05 eV 1.7 1.75 1.8 1.85 1.9 1.95 1000/T (K–1)

1.65

1.34

1.41

1.66

1.46 1.0

Fig. 9.25

Migration at periphery of Ir18 on Ir(111) surface derived from MC studies. (a) Solid curve shows energy for single atom peripheral diffusion. Dashed curve gives effective energy for dimer process. Numerals below a configuration indicate percentage of time spent there. (b) Peripheral diffusion of dimer around Ir18. (c) Arrhenius plot for the diffusivity of Ir18 derived from mean-square displacements. Solid circles: from Monte Carlo simulations. Open triangles: experiments of Wang et al. [34] (after Zhuang and Wang [43]). (b)

(c)

[112]

(a)

[110] Fig. 9.26

Diffusion of Pt11 cluster on Pt(111) by reptation, based on EAM. (a) Starting position. (b) Upper part of cluster glides, forming stacking fault. (c) Form after diffusion (after Chirita et al. [44]).

is also proposed to be important for close-packed clusters of 19 atoms. The value obtained for the heptamer is essentially identical with experiments [47], but that derived for Pt19 appear too high. Diffusion of islands on Cu(111) and Ag(111) was extensively studied with scanning tunneling microscopy by Schlösser et al. [48], who examined the relation between the

689

9.2 Large clusters on fcc(111) surfaces

4

1 3

2 –3

1.5 nm 2.25 nm 3.65 nm 6.1 nm

In D

–4

core

B EA cb , E cb(c)

–5 E ku

–7

B EA cb , E cb(s)

(a)

–6

a E kd

E = 0.48 ± 0.06 ± 0.04 eV (errors: data selection - statistics)

2.9

3.0

3.1 1000/T (K–1)

Fig. 9.27

1: ED 2: E A 3: EB 4: Ecd

3.2

(b)

A /B E cc

Diffusivity of vacancy islands on Cu(111) derived from STM measurements. (a) Arrhenius plots. (b) Atomic steps in vacancy island motion (after Schlösser et al. [48]).

diffusivity and the island radius Rr. Measurements were made at temperatures from 263 K to 343 K on islands with radii in the range from 1 to 10 nm. For vacancy islands on L for the size dependence was given Cu(111) the exponent αL in the expression D / R
α r as 1.37 ± 0.1, and the activation energy for island diffusion derived from four temperatures, shown in Fig. 9.27a, amounted to 0.49 ± 0.01 eV. For similar vacancy islands on Ag(111) the exponent αL was 1.33 ± 0.11 and the diffusion barrier came to 0.51 ± 0.04 eV. With adatom islands on Ag(111), αL was 1.63 ± 0.38, and the diffusion barrier amounted to 0.53 ± 0.05 eV. The activation energies for all three systems were much the same. What is most significant, however, is that the exponent αL in the size relationship for all three differed significantly from the value of three expected for peripheral diffusion of atoms around the island edges as the limiting step. Calculations of the activation energies were made with EMT potentials for the various atomic processes in island diffusion. The processes taken into account are illustrated in Fig. 9.27b, the energetics derived from calculations are listed in Table 9.5. However, Schlösser et al. ended their work with the conclusion that “a simple interpretation of island diffusion in terms of a single rate-limiting process is not possible.” The authors concluded that island diffusion arose from a combination of peripheral motion of single atoms plus the breakup of island cores. Monte Carlo simulations of copper island diffusion have been done both on Cu(100) and Cu(111) surfaces by Jahma et al. [19], based on energetics from effective medium theory. For islands on Cu(111) the diffusivity versus cluster size curves are smooth with no oscillations. There is also relatively little temperature dependence of the diffusivity for small islands, and

690

Diffusion of large clusters

Table 9.5 Energetics of cluster self-diffusion on (111) surfaces in eV, calculated using EMT potentials [48]. Cu(111)

Ag(111)

γS ΔEks ΔEcs ΔEkt ED EA EB Ecd A=B Ekd Eku EA cb EBcbðcÞ EBcbðsÞ

0.211 0.223 0.434 0.721 0.057 0.225 0.325 0.370 EA=B þ DEks Ecd + ΔEks EB þ DEcs EA þ DEcs Ecb + ΔEcs

0.159 [49] 0.169 0.330 0.558 0.068 0.221 0.300 0.318

EA=B cc

EA=B þ 2DEks

γS – formation energy per step atom; ΔEks, ΔEcs and ΔEkt – binding energy of kink atom, of core atom relative to adatom at the step, and of the kink atom relative to the terrace. The diffusion energies in the table are relative to the atomic motions shown in Fig. 9.27b.

the activation energy for single adatom diffusion was 0.026 eV. For islands made of 19 to 100 atoms the exponent αN in the size relation was about 1.57, in agreement with the view that large island movement is dominated by diffusion along the perimeter. Recently Ghosh et al. [50] have carried through a quite thorough Monte Carlo simulation of cluster diffusion on Cu(111) with the NEB method, using EAM interaction potentials and relying on a pattern recognition scheme to identify 49 different diffusion processes. The energetics of the main processes are shown in Table 9.6 for a cluster of 10 atoms. General clusters of 10, 18, and 26 atoms were found to carry out overall random motion. Compact hexagonal clusters, with 19 or 37 atoms, diffused slowly, with occasional large transitions, as shown in Fig. 9.28b. The authors also found another two classes of clusters: with 6s+3 atoms, s being an integer and with a hexagon+1. These were found to execute stick-slip motion, with short transitions between long, essentially stationary periods. At higher temperatures, clusters of all sizes were moving randomly, and vacancies started to form. Arrhenius plots for the diffusivity of general clusters are shown in Fig. 9.28d, and yield activation energies reasonably close together but increasing with size. The most frequent event in these clusters is diffusion along A-type edges, with B-type edges next. Effective barriers for clusters, except for those of 21 and 26 atoms, turned out to be around 0.65 eV. For a cluster made up of 21 atoms the effective barrier for movement of the center of mass proved to be higher, 0.84 eV. For clusters composed of 26 atoms the effective activation energy was derived as 0.76 eV. Hexagonal compact entities diffused so slowly that no barrier heights were evaluated, which is regrettable, as comparison with the

691

9.2 Large clusters on fcc(111) surfaces

Table 9.6 Energetics based on EAM for main processes in diffusion of 10 atom cluster on Cu(111) [50].

Process

Energy barrier Process (eV)

Energy barrier (eV)

Step edge A Step edge B Kink detach along step A Kink detach along step B Kink incorporation step A Kink incorporation step B Kink detach out of step A Kink detach out of step B Kink fall into step A Kink fall into step B Kink rounding at step A Kink rounding at step B Reverse of step rounding A

0.252 0.295 0.519 0.556 0.220 0.265 0.658 0.590 0.074 0.007 0.656 0.678 0.374

0.402 0.313 0.143 0.010 0.374 0.038 0.052 0.317 0.084 0.396 0.015 0.063 0.019

6 (b)

(a) y coordinate (Å)

y coordinate (Å)

0

Reverse of step rounding B Corner rounding AA at stage 1 Corner rounding AA at stage 2 Corner rounding AA at stage 3 Corner rounding BB at stage 1 Corner rounding BB at stage 2 Corner rounding BB at stage 3 Corner rounding AB at stage 1 Corner rounding AB at stage 2 Corner rounding BA at stage 1 Corner rounding BA at stage 2 Rounding chain A Rounding chain B

–200 –400 –600 –800 –400

–300

0 –200 –100 x coordinate (Å)

100

4

2

0

–2 –2

200

0

2 4 x coordinate (Å)

6

30 10 atoms; Eeff = 0.64 eV 14 atoms; Eeff = 0.66 eV 18 atoms; Eeff = 0.65 eV 26 atoms; Eeff = 0.71 eV

20 0

In D

y coordinate (Å)

5

–5

10 0

(c)

(d) 0

Fig. 9.28

5 10 x coordinate (Å)

15

20

–10 10

20

30

40

1/kT (eV –1)

Trace of center-of-mass diffusion obtained from MC simulations using EAM of clusters on Cu (111). (a) Cu10 (b) Cu19 (c) Cu21 (d) Arrhenius plots for diffusion derived from Monte Carlo simulations (after Ghosh et al. [50]).

experiments of Wang et al. [34] would have been interesting. However, there clearly is a similarity in behavior of hexagonal clusters on Ir(111) and Cu(111). From the information presented for islands of more than 10 atoms on an fcc(111) surface, we can conclude that movement of most clusters proceeds by adatom edge

692

Diffusion of large clusters

Fig. 9.29

Diffusion of Ni clusters on Au(110)-(1 × 2) simulated with Johnson potentials [46,52]. Illustration of rolling in trough for (a) Ni4 (b) Ni7 (c) Ni13 (after Fan and Gong [51]).

running. For some systems, for example like Al(111) [32], movement along edges by exchange should be taken into account. The evaporation–condensation mechanism has been excluded from the picture at least at room temperature. Investigations at higher temperatures are still in order. Movement of hexagonal islands is rather different. The mechanism of movement as well as the reasons for the high prefactors for diffusivity and for the presence of long jumps remain to be assigned with certainty. The small seven atom hexagonal clusters may move by gliding; the mechanism of movement for bigger hexagonal islands is still uncertain. Does it proceed by creation of a dislocation or by shuffle gliding? Individual atom-by-atom movement is certainly unable to explain the observed displacements, and concerted motion should play a role. Unfortunately experimental data about movement of large clusters on fcc(111) surfaces is limited to three materials: iridium, copper, and silver.

9.3

Large clusters on fcc(110) surfaces Information about the movement of large clusters on the channeled fcc(110) surface is scarce. For metallic clusters the only study found was for NiN on the Au(110)-(1 × 2) reconstructed surface, done by Fan and Gong [51] with EAM and the interactions of Johnson [46,52]. Since Ni-Ni interactions are stronger than Au-Ni, the Ni13 cluster prefers the three-dimensional structure of an icosahedron. Such a cluster can roll in the deep channels of the Au surface, shown in Fig. 9.29. A sphere-like shape is also achieved for Ni4 and Ni7 clusters. The diffusion constants are orders of magnitude larger for clusters with spherical structure, compared to clusters which cannot achieve this form. In rolling, only two atoms need overcome an energy barrier. Although rolling seems to be the leading mechanism, gliding and rotation was also observed. It is of interest to note diffusion studies on rather different materials, decacyclene, and hexa-tert-butyl decacyclene [53]. These flat ring structures have been studied on Cu(110). They are mentioned here because just as in Ir19 [34], long jumps as large as 3.9 ± 0.2 and 6.8 ± 0.3 copper nearest-neighbor spacings have been found for the two molecules, with prefactors of 10−1.0 ± 1.0 and 100.9 ± 1.0 cm2/sec. These are again unusually large, and it appears that long jumps may be quite common in the diffusion of such large entities. Experimental mechanistic studies of large clusters on channeled surfaces have been overlooked, and we have no real clue about the mechanisms of movement or the diffusivities.

References

9.4

693

Comments and comparisons What is clear from this brief survey is that much progress has been made in understanding the diffusion of large clusters over crystal surfaces. Evaporation–condensation does not appear to be a significant process in surface diffusion under the usual conditions. Instead it seems that peripheral diffusion of atoms around the cluster edges is the dominant mechanism. More could be done to define the mechanism of motion for hexagonal clusters with edges of equal length. It is clear, however, that such clusters have unusually large diffusion prefactors, and surprisingly, carry out jumps large by comparison with nearest-neighbor surface spacings. It is evident, moreover, that more quantitative experimental studies will be really desirable.

References [1] G. Ehrlich, in: Proceedings of the 9th International Conference on Solid Surfaces, J. L. de Segovia (ed.), Layer growth – an atomic picture, (A.S.E.V., Madrid, 1983), p. 3–16. [2] G. Ehrlich, in: Chemistry and Physics of Solid Surfaces V, R. Vanselow, R. Howe (eds.), An atomic view of crystal growth, (Springer-Verlag, Berlin, 1984), p. 282–296. [3] H.-W. Fink, in: Diffusion at Interfaces – Microscopic Concepts, M. Grunze, H. J. Kreuzer, J. J. Weimer (eds.), Direct observation of atomic motion on surfaces, (Springer-Verlag, Berlin, 1988), p. 75–91. [4] K. Binder, M. H. Kalos, “Critical clusters” in a supersaturated vapor: Theory and Monte Carlo simulation, J. Stat. Phys. 22 (1980) 363–396. [5] A. F. Voter, Classically exact overlayer dynamics: Diffusion of rhodium clusters on Rh(100), Phys. Rev. B 34 (1986) 6819–6829. [6] A. F. Voter, Simulation of the layer-growth dynamics in silver films: Dynamics of adatom and vacancy clusters on Ag(100), Proc. SPIE 821 (1987) 214–226. [7] J.-M. Wen, S.-L. Chang, J. W. Burnett, J. W. Evans, P. A. Thiel, Diffusion of large twodimensional Ag clusters on Ag(100), Phys. Rev. Lett. 73 (1994) 2591–2594. [8] W. W. Pai, A. K. Swan, Z. Zhang, J. F. Wendelken, Island diffusion and coarsening on metal (100) surfaces, Phys. Rev. Lett. 79 (1997) 3210–3213. [9] J. M. Wen, J. W. Evans, M. C. Bartelt, J. W. Burnett, P. A. Thiel, Coarsening mechanisms in a metal film: From cluster diffusion to vacancy ripening, Phys. Rev. Lett. 76 (1996) 652–655. [10] C. R. Stoldt, C. J. Jenks, P. A. Thiel, A. M. Cadilhe, J. W. Evans, Smoluchowski ripening of Ag islands on Ag(100), J. Chem. Phys. 111 (1999) 5157–5166. [11] S. Pal, K. A. Fichthorn, Size dependence of the diffusion coefficient for large adsorbed clusters, Phys. Rev. B 60 (1999) 7804–7807. [12] A. Lo, R. T. Skodje, Diffusion and evaporation kinetics of large islands and vacancies on surfaces, J. Chem. Phys. 111 (1999) 2726–2734. [13] J. Heinonen, I. Koponen, J. Merikoski, T. Ala-Nissila, Island diffusion on metal fcc (100) surfaces, Phys. Rev. Lett. 82 (1999) 2733–2736. [14] J. Merikoski, T. Ala-Nissila, Diffusion processes and growth on stepped metal surfaces, Phys. Rev. B 52 (1995) R8715–8718.

694

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[15] G. Mills, T. R. Mattsson, L. Mollnitz, H. Metiu, Simulations of mobility and evaporation rate of adsorbate islands on solid surfaces, J. Chem. Phys. 111 (1999) 8639–8650. [16] W. W. Pai, J. F. Wendelken, C. R. Stoldt, P. A. Thiel, J. W. Evans, D.-J. Liu, Evolution of two-dimensional wormlike nanoclusters on metal surfaces, Phys. Rev. Lett. 86 (2001) 3088–3091. [17] F. J. Resende, B. V. Costa, Molecular dynamics study of copper cluster deposition on a Cu(010) surface, Surf. Sci. 481 (2001) 54–66. [18] X. Wang, F. Xie, Q. Shi, T. Zhao, Effect of atomic diagonal motion on cluster diffusion coefficient and its scaling behavior, Surf. Sci. 561 (2004) 25–32. [19] M. O. Jahma, M. Rusanen, A. Karim, I. T. Koponen, T. Ala-Nissila, T. S. Rahman, Diffusion and submonolayer island growth during hyperthermal deposition on Cu(100) and Cu(111), Surf. Sci. 598 (2005) 246–252. [20] J. R. Sa´nchez, Metal surface adsorbed clusters: Structure and dynamics, J. Molec. Catalysis A: Chemical 237 (2005) 206–209. [21] S. C. Wang, G. Ehrlich, Diffusion of large surface clusters: Direct observations on Ir(111), Phys. Rev. Lett. 79 (1997) 4234–4237. [22] D. J. Trevor, C. E. D. Chidsey, Room temperature surface diffusion mechanisms observed by scanning tunneling microscopy, J. Vac. Sci. Technol. B 9 (1991) 964–968. [23] J. de la Figuera, J. E. Prieto, C. Ocal, R. Miranda, Creation and motion of vacancy islands on solid surfaces: A direct view, Solid State Comm. 89 (1994) 815–818. [24] K. Morgenstern, G. Rosenfeld, B. Poelsema, G. Comsa, Brownian motion of vacancy islands on Ag(111), Phys. Rev. Lett. 74 (1995) 2058–2061. [25] G. Rosenfeld, K. Morgenstern, G. Comsa, in: NATO-ASI Surface Diffusion: Atomistic and Collective Processes, M. C. Tringides (ed.), Diffusion and stability of large clusters on crystal surfaces, (Plenum Press, New York, 1997), p. 361–375. [26] J. C. Hamilton, M. S. Daw, S. M. Foiles, Dislocation mechanism for island diffusion on fcc(111) surfaces, Phys. Rev. Lett. 74 (1995) 2760–2763. [27] J. C. Hamilton, Magic size effects for heteroepitaxial island diffusion, Phys. Rev. Lett. 77 (1996) 885–888. [28] D. S. Sholl, R. T. Skodje, Diffusion of clusters of atoms and vacancies on surfaces and the dynamics of diffusion-driven coarsening, Phys. Rev. Lett. 75 (1995) 3158–3161. [29] L. Bitar, P. A. Serena, P. García-Mochales, N. García, V. T. Binh, Mechanism for diffusion of nanostructures and mesoscopic objects on surfaces, Surf. Sci. 339 (1995) 221–232. [30] L. Kuipers, M. S. Hoogeman, J. W. M. Frenken, Step dynamics on Au(110) studied with high temperature, high speed scanning tunneling microscope, Phys. Rev. Lett. 71 (1993) 3517. [31] A. Bogicevic, S. Liu, J. Jacobsen, B. Lundqvist, H. Metiu, Island migration caused by the motion of the atoms at the border: Size and temperature dependence of the diffusion coefficient, Phys. Rev. B 57 (1998) R9459–9462. [32] A. Bogicevic, J. Strömquist, B. I. Lundqvist, Low-symmetry diffusion barriers in homoepitaxial growth of Al(111), Phys. Rev. Lett. 81 (1998) 637–640. [33] S. C. Wang, G. Ehrlich, Equilibrium shapes and energetics of iridium clusters on Ir(111), Surf. Sci. 391 (1997) 89–100. [34] S. C. Wang, U. Kürpick, G. Ehrlich, Surface diffusion of compact and other clusters: Irx on Ir(111), Phys. Rev. Lett. 81 (1998) 4923–4926. [35] S. Y. Krylov, Surface gliding of large low-dimensional clusters, Phys. Rev. Lett. 83 (1999) 4602–4605.

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[36] J. C. Hamilton, A. F. Voter, Failure of 1D models for Ir island diffusion on Ir(111), Phys. Rev. Lett. 85 (2000) 1580. [37] G. H. Vineyard, Frequency factors and isotope effects in solid state rate processes, J. Phys. Chem. Solids 3 (1957) 121–127. [38] S. Y. Krylov, Krylov replies, Phys. Rev. B 85 (2000) 1581. [39] U. Kürpick, B. Fricke, G. Ehrlich, Diffusion mechanisms of compact surface clusters: Ir7 on Ir (111), Surf. Sci. 470 (2000) L45–51. [40] J. C. Hamilton, M. R. Sorensen, A. F. Voter, Compact surface-cluster diffusion by concerted rotation and translation, Phys. Rev. B 61 (2000) R5125–5128. [41] A. P. Sutton, J. Chen, Long-range Finnis-Sinclair potentials, Philos. Mag. Lett. 61 (1990) 139–46. [42] S. P. Chen, Studies of iridium surfaces and grain boundaries, Philos. Mag. A 66 (1992) 1–10. [43] G. Zhuang, W. Wang, The atomic moving process of cluster Ir18 diffusion on Ir(111), Internatl. J. Mod. Phys. B 14 (2000) 427–434. [44] V. Chirita, E. P. Münger, J. E. Greene, J.-E. Sundgren, Cluster diffusion and surface morphological transitions on Pt(111) via reptation and concerted motion, Thin Solid Films 370 (2000) 179–185. [45] D. J. Oh, R. A. Johnson, Simple embedded atom method model for fcc and hcp metals, J. Mater. Res. 3 (1988) 471–478. [46] R. A. Johnson, Alloy models with the embedded atom method, Phys. Rev. B 39 (1989) 12554–12559. [47] K. Kyuno, G. Ehrlich, Diffusion and dissociation of platinum clusters on Pt(111), Surf. Sci. 437 (1999) 29–37. [48] D. C. Schlösser, K. Morgenstern, L. K. Verheij, G. Rosenfeld, F. Besenbacher, G. Comsa, Kinetics of island diffusion on Cu(111) and Ag(111) studied with variable-temperature STM, Surf. Sci. 465 (2000) 19–39. [49] P. Stoltze, Simulation of surface defects, J. Phys.: Condens. Matter 6 (1994) 9495–9517. [50] C. Ghosh, A. Kara, T. S. Rahman, Usage of pattern recognition scheme in kinetic Monte Carlo simulations: Application to cluster diffusion on Cu(111), Surf. Sci. 601 (2007) 3159–3168. [51] W. Fan, X. G. Gong, Simulation of Ni cluster diffusion on Au(110)-(1 × 2) surface, Appl. Surf. Sci. 219 (2003) 117–122. [52] R. A. Johnson, Analytical nearest-neighbor model for fcc metals, Phys. Rev. B 37 (1988) 3924–3931. [53] M. Schunack, T. R. Linderoth, F. Rosei, E. Laegsgaard, I. Stensgaard, F. Besenbacher, Long jumps in the surface diffusion of large molecules, Phys. Rev. Lett. 88 (2002) 156102 1–4.

10 Atomic pair interactions

As is clear from the previous chapters, a considerable amount of information has become available about the diffusivity of metal atoms on metal surfaces. This knowledge will have an impact on understanding topics such as crystal growth and dissolution, annealing, sintering, as well as chemical surface reactions. We will not examine any of these important topics, but will instead focus on attempts to measure atomic surface interactions, as an example of effects which have become much clearer through gains in the knowledge of surface diffusion. Interactions can be divided into two categories, direct, for example van der Waals, dipolar, as well as electronic interactions, and indirect, which are mediated by the substrate, such as coupling by electronic states, elastic effects, or vibrational coupling. In Chapters 8 and 9 we described the movement of clusters created mostly by direct interactions of adatoms. In this chapter we will concentrate on the second type of interactions, mediated by the lattice – indirect interactions. The theory of interactions between adsorbed atoms has been nicely reviewed by Einstein [1], so here we just want to point out that with two atoms in the gas phase, the wave function for a higher state will be confined to the vicinity of the core. Placed on a metal surface, however, quantum interference of wave functions enters and adatom waves combine with metal wave functions from the lattice. The contributions from two atoms may overlap, as shown in Fig. 10.1 [2]. When in phase, attraction occurs between the two atoms, when out of phase, there will be repulsion. In view of the oscillatory nature shown in the figure it is clear that there will also be oscillations in the interactions as the interatomic separation is varied. The first piece of information of interest in surface interactions is the energy of two adsorbed atoms as a function of their separation. What is the range of these effects? How strong are they? A variety of techniques have been used in the past in an effort to gain an insight into these questions. Such efforts have been briefly reviewed some time ago [3]. With modern methods, difficulties in the measurements are not hard to overcome. What needs to be done is to place two atoms on a surface, and allow them to equilibrate with each other. This is easy to do in the STM or FIM, as two atoms can be directly observed. How the atoms position themselves will be dependent upon interactions between the adatoms on the surface. Crucial is knowledge about the diffusion characteristics of the atoms, as conditions have to be adjusted to ensure that equilibrium is established. The analysis of the distribution of atom separations to give interactions is straightforward, and a simple derivation [4] proceeds as follows.

697

Atomic pair interactions

Fig. 10.1

Wave functions for two atoms (a) in the gas phase and (b) adsorbed on a metal surface as a function of separation. For the latter there are interactions between the two adatoms (after Grimley [2]).

Suppose there are two equilibrated atoms located at sites i and j on the surface, and the atoms are in a quantum state ‘. The probability Pij;‘ of finding these two atoms is given by X Pij;‘ ¼ exp½
Eij;‘ =kT= exp½
Eij;‘ =kT ¼ exp½
Eij;‘ =kT=Z; (10:1) ij;‘

here Eij;‘ is just the energy of the two atoms in state ‘, and Z is the canonical partition function X exp½
Eij;‘ =kT: (10:2) Z¼ ij;‘

Only the spatial distribution of the atoms will be measured. Summing over all states ‘ we therefore obtain X Pij ¼ exp½
Eij;‘ =kT=Z: (10:3) ‘

Now the Helmholtz free energy Fij can be written as X
Fij =kT ¼ ln exp½
Eij;‘ =kT;

(10:4)

‘

so that the probability Pij of finding a pair at sites i and j is just Pij ¼ exp½
Fij =kT=Z:

(10:5)

Our real interest, of course, is not the probability Pij, but rather the probability P(R) of finding atoms at a vector separation R on the surface, for which the total number of pairs possible at this separation is No (R). This we obtain by summing Eq. (10.5) over all pairs of sites separated by R PðRÞ ¼ CNo ðRÞ exp½
FðRÞ=kT;

(10:6)

698

Atomic pair interactions

Here C is a constant introduced to give normalization, and includes the partition function Z. The probability P(R) is given in terms of N(R), the number of observations for which a pair of atoms has been found separated by R, that is PðRÞ ¼ NðRÞ=M;

(10:7)

where M is the total number of observations, so that NðRÞ ¼ CMNo ðRÞ exp½
FðRÞ=kT:

(10:8)

We can therefore write the free energy of interaction F(R) as FðRÞ=kT ¼ C1
ln½NðRÞ=No ðRÞ;

(10:9)

where we have written C1 for exp(CM). This new normalization can be obtained by summing over all separations larger than a critical value Rc for which interactions have vanished, so that " # X X C1 ¼ ln NðRÞ= No ðRÞ : (10:10) R4Rc

R4Rc

Gaining an understanding of how the free energy of interaction between two atoms varies with their separation R is now just a matter of measuring N(R), the number of times a pair is observed separated by R in an equilibrated system. Such measurements can be made by either field ion microscopy or with the scanning tunneling microscope; the latter technique has only been utilized recently, but most successfully.

10.1

Early measurements In 1973, measurements of the spatial distribution and of the interactions between rhenium atoms on the (110) plane of tungsten were published by Tsong [5]. It was noticed that two Re atoms migrated over the surface, maintaining a distance of 6–7.5Å, implying that the potential should have two minima at 2.74 and 6.8 Å. The results of more detailed analyses shown in Fig. 10.2 were interesting, but not really correct. Observations were made with five rhenium atoms on a plane, rather than just two, and the statistics were poor. Sites on the surface were not established; instead, observations were analyzed just according to the distance between the atoms, which is not reliable as the magnification of the FIM changes across the plane. Despite these defects, what is worth noting here is a strong attractive interaction, at ~2.5 Å, and an interaction potential that seems to vary between maxima and minima as the distance increases. The latter is, of course, what is expected for Friedel oscillations [6]. It must be noted, however, that no such oscillations were found for tungsten atoms, that conditions for establishing equilibrium were not stated, and the anisotropy of the surface was not considered. A year later, a more careful study appeared of two tungsten atoms diffusing on the (211) plane of tungsten, separated from each other by an empty channel, as in Fig. 10.3a [7]. For this system, the probability of finding two atoms separated by the distance X along the channels is

699

10.1 Early measurements

(b)

25

–100 Re-Re INTERACTION T = 330 K

20 15 10 5 0

0

Fig. 10.2

5

10 15 Separation (Å)

20

25

Cohensive energy (meV)

Number of events

(a)

–50

0

5

10

20

25

Distance (Å)

50 160 100

Results obtained from FIM observations of five Re atoms on W(110) plane. (a) Distribution of pair separations at 300 K. (b) Interaction energies between adatoms, derived from (a), as a function of their separation (after Tsong [5]).

pðXÞ ¼ Cð2
δx0 ÞðL
XÞ exp½
FðXÞ=kT;

(10:11)

where L is the number of sites in a channel and δx0 is the Kronecker delta. Tungsten atoms on W(211) always diffuse in the same channel, so that such measurements are easy. The distribution, shown in Fig. 10.3c, does not differ significantly from that of two non-interacting particles, although there may be a slight diminution when the X-separation is zero. The distance between the two rows in these experiments was 8.95 Å. Measurements were also done on the distribution of separations between two tungsten atoms in adjacent rows of W(321), separated by 7.08 Å. Again no interactions were found, but it should be noted that only 300 observations were made, which limits the accuracy. However, two atoms in adjacent channels on W(211), at a distance of 4.48 Å, interact strongly with each other. This study showed that for a pair of W atoms on a bcc channelled surface interactions across the channels become significant once the spacing is shorter than 7Å. Long-range interactions for tungsten atoms in adjacent channels of W(321) were suggested by Nishigaki and Nakamura [8] but this was based on observations of only 10 pairs, and therefore cannot be considered definitive. More qualitative but also more pleasing experiments were reported by Ayrault [9], who measured the mean-square separation between two rhodium atoms diffusing in adjacent channels of Rh(110), L spacings long. If the atoms are independent of each other, then if atom 1 is held at a given site while 2 moves randomly in the second row, D E 1 ZL i 1 h 2 ðx2
x1 Þ ¼ ðx2
x1 Þ2 dx2 ¼ ðL
x1 Þ3 þx31 : L 3L

(10:12)

0

However, for atom 1, all positions in its channel are really equally likely, so that the mean-square separation is obtained by averaging over all accessible values of x1, to get D E ðx2
x1 Þ2 ¼ L2 =6: (10:13)

700

Atomic pair interactions

-[111] [011]

(b) (a) X

40

Simulation Experiment

Number of observations

30 20 10 0 40

Predicted Experiment

30 20 10 0

(c) Fig. 10.3

0 2 4 6 8 10 12 14 Pair separation X along channels

Distribution of distances between two tungsten adatoms at 295 K in rows of W(211) plane separated by an empty channel, as in (a), giving a minimum separation of 8.95 Å [7]. (b) FIM image of two W atoms in rows separated by an empty one on W(211). Arrow gives direction of motion. (c) Distribution of distances between W adatoms. No significant deviations from random distribution are evident.

When there are interactions between the two atoms, the mean-square separation was written as D E ðx2
x1 Þ2 ¼ L2 =6Ai ; (10:14) where attractive forces give Ai > 1 and repulsive ones Ai < 1. Results obtained on the (110) plane as well as on (331) are listed in Table 10.1. It is clear that there are strong attractive interactions between atoms in adjacent channels, and for Rh adatoms they seem to persist even when the atoms are separated by an adjacent channel. This suggests that pair interactions of Rh across channels of the fcc(110) surface are of longer range than for W pairs. Unfortunately, no further energy estimates were made.

701

10.1 Early measurements

Table 10.1 Correlated Rh atom motion on channeled rhodium planes [9]. Plane

Channel spacing (Å)

Closest Approach (Å)

Observations

Ai

T (K)

(110) (110) (331)

3.80 3.80 5.86

7.61 3.80 11.7

18 47 30

6.0 23 1.7

176–198 190–215 216

(a)

(b) 343 K

1.0

1.0 0.5

0.5

1 nm

1 nm 0

Fig. 10.4

360 K

1.5 g(R)

g(R)

1.5

0.25

0.5 IRI/dR

0.75

1.0

0

0.25

0.5 IRI/dR

0.75

1.0

Pair distribution g(R) = Ρ(R)/2π|R| for Re adatoms on W(110) plane, observed in FIM. Distributions equilibrated at two temperatures (a) 343 K and (b) 360 K. Dashed lines give random distribution on structureless plane. Differences suggest that 343 K was too low a temperature for adequate equilibration; dR denotes plane diameter (after Bassett and Tice [10]).

Previous estimates of the pair distribution of rhenium on W(110) made by Tsong [5] were criticized in 1975 by Bassett and Tice [10], whose own determinations are shown in Fig. 10.4. Noticeable here is the absence of a strong attraction at small separations, which had been reported by Tsong. Furthermore, they found a significant difference in the distribution on equilibrating at a higher temperature compared with a 20 K lower one, which they suggested showed poor equilibration at the lower temperature of 343 K. They concluded that “the evidence that U(r) for rhenium adatoms on W(110) has the oscillatory character associated with indirect coupling” “is certainly not conclusive”. However, the observations were indicative of repulsion at separations less than 5 Å and weak attractions between 7 Å and 10 Å. Some years later, Fink et al. [11] did quantitative studies with a palladium atom together with a tungsten atom, and later also with a rhenium atom, on W(110). Palladium diffuses at temperatures below 250 K, at which the tungsten and also the rhenium adatom remain stationary, giving a reliable reference point on the surface and confining the atoms to the center of the plane. As seen in the distribution in Fig. 10.5a, consisting of 110 annealing steps at 240 K, palladium most frequently stays at a separation of ~3 Å, the next most frequent position was observed at 11 Å, and possibly a third maximum at 18 Å. For the smaller separation, the atoms were located along the directions, and no atoms were observed between the first and second population peaks. Results obtained for Re-Pd in 51 diffusion intervals at 250 K are shown in Fig. 10.5c. Occupation of nearest-neighbor sites is prevented by repulsive interactions, but there are

702

Atomic pair interactions

Fig. 10.5

Observations of W-Pd and Re-Pd pairs on W(110) plane in FIM after equilibration. (a) Location of Pd adatoms after diffusion around W adatom at 240 K. (b) Distribution of Pd around W in its dependence upon separation jRj at 240 K. (c) Distribution of separations between stationary Re adatom and Pd, after diffusion at 250 K. Attractive interaction at 5 Å is evident (after Fink et al. [11]).

attractive interactions at 5 Å along . Another separation of 8 Å was not as strongly preferred. From this study it emerged that the direction is preferred for bonding of a Re-Pd pair, in close-bonded dimers as well as for larger interatomic distances. This paper really provided the first experimental indication of long-range interactions. Attempts to arrive at better evaluations of long-range interactions on the (110) plane of tungsten were again taken up by Tsong and Casanova [12,13]. They examined effects for Re-Re, W-Ir, W-Re, and Ir-Ir with two atoms on the plane, but still maintained a radial analysis rather than endeavoring to identify the binding sites. However, the analysis was improved in that the (110) plane was now assumed to be elliptical [13]. The pair energy for interactions between two iridium atoms is plotted in Fig. 10.6a. There is an attractive minimum at ~ 4 Å. At larger distances, on the order of 7 Å, there is a clear repulsion between the atoms; at bigger separations there is an attraction, followed by what look like weak oscillations. The interactions for W-Ir are shown in Fig. 10.6b. This diatomic cluster is less stable than Ir-Ir. The pair distribution shows an attractive region around 3 Å, followed by repulsion around 5 Å, and another attraction around 12 Å. Pair energies for Re-Re and Re-W are plotted in Fig. 10.6c and d. During equilibration at 400 K, closely bonded, unstable Re-Re dimers were occasionally created. From their dissociation it was concluded that around 3 Å the cohesive energy was ~260 meV. However, such dimers were not observed in 1045 heating periods of 60 sec at 330 K. The closest interatomic distance found was 6 Å. Based on these observations, the authors concluded that atoms repelled each other up to ~5 Å, and bonded weakly around 7 Å. The situation changed for W-Re dimers, for which observations of closely bonded dimers were frequent. Out of 1111

703

10.1 Early measurements

(a) Ir-Ir

(b) W-Ir 40

0

40 10

20 30 Distance (Å)

20

40 30 Distance (Å)

20

40 30 Distance (Å)

10

–60

–100

Pair-energy (meV)

0

–40

–80

(c) Re-Re

40

60 40

(d) W-Re

20 0

10

–20

20 0

20

–20

–20

80

Pair-energy (meV)

20

Pair-energy (meV)

Pair-energy (meV)

40

10

20

30

40

–60

Distance (Å) –20

–80 –100

Fig. 10.6

Interaction between two adatoms on W(110) plane, deduced from distance distribution observed with FIM. (a) Ir-Ir pair after diffusion at 330 K. Repulsion is apparent at 7 Å. (b) Interaction energies between W and Ir adatom at different separations on W(110), after diffusion at 300 K. Strong repulsion is seen at ~5 Å. (c) Re-Re energies derived from distance distribution. (d) W-Re interactions (after Tsong and Casanova [13]).

observations, the close dimer was observed as many as 332 times. The repulsive region was between 3 and 6 Å, with an energy around 50 meV. All the systems studied, except for Re-Re, had a strong attractive potential at a nearest-neighbor spacing, indicating interactions of − 99.0 ± 0.7 meV for W-Re, − 82.0 ± 2.5 for Ir-Ir, and − 53.2 ± 3.6 for W-Ir. For W-Re, there is repulsion between 3 and 6 Å, with possible oscillations at higher spacings. Fairly clear oscillations are visible for W-Ir, but were not that well defined for Ir-Ir. The problem with these determinations is that, although the number of observations had been increased to ~1000, no attempt was made to relate interactions to the location of sites on the surface. In 1983 Stoop [14] investigated pairwise interactions for describing the phase diagram of a silver submonolayer on the W(110) surface. From his work it is clear that pairwise

704

Atomic pair interactions

Table 10.2 Interaction energies (eV) on W(110). ε1, ε2, ε3 – first, second, and third nearest-neigbor pairwise interactions, ξ1, ξ2 – trio interactions, ς – quarto interactions.

Ag Cu Au Ag

ε1

ε2

ε3

ξ1

ξ2

− 0.074 − 0.222 − 0.293 − 0.177

0.037 − 0.147 − 0.255 − 0.167

0.037 − 0.072 − 0.146 − 0.103

− 0.037 0.008 0.023 0.007

− 0.037 0.052 0.158 0.094

ς

Reference

− 0.148 − 0.085

[14] [16] [16] [3]

interactions alone are not enough to describe the experimental phase diagram. He obtained the best fit using attractive first nearest-neighbors, repulsive second and third nearest-neighbor interactions, together with two types of attractive three-body effects. Pairwise interactions extended out to fifth nearest-neighbor distances. This effort was continued by Roelofs and Bellon [15] for Cu and Au based on the cluster calculations of Gollish [16]; their findings for pair, trio and quarto interactions are listen in Table 10.2. Ehrlich and Watanabe [3] followed Roelofs’ and Bellon’s procedure and derived values for Ag on W(110), also shown in Table 10.2. All pair interactions were attractive and trios repulsive. Even if the data do not agree with the previous findings of Stoop [14], it reproduced the general trends of work functions from experiments by Kolaczkiewicz and Bauer [17]. It was clear that more detailed investigations were needed.

10.2

More recent studies A start on more careful studies of long-range atomic interactions was made by Fink [18], who mapped out the sites on the W(110) surface at which atoms were held. This was accomplished by allowing a rhenium atom to diffuse over the surface at 380 K, and determining its position after each of 345 diffusion intervals. The grid of binding sites so established is shown in Fig. 2.17, and provides a sound basis for finding distances between two adatoms on the plane; temperature was adjusted so that the root-meansquare displacement of an atom was comparable to the diameter of the plane. The distribution of separations found between two rhenium atoms on W(110) equilibrated at 400 K is given in Fig. 10.7a, and is compared with the distribution for two noninteracting particles obtained by simulation. The statistics are not great, with only 381 observations, but what clearly emerges again are repulsive interactions at separations less than 10 Å. No attempt to evaluate effects at longer distances was made. Quite surprising here is the lack of dimer formation with two Re atoms, even at low temperatures where atom mobility is reduced. Based on this data we expect the same behavior for the rhenium trimer Re3. Surprisingly enough, this was not the case – stable trimers were formed, as shown in Fig. 10.7b. Clearly three-body interactions are important, with a binding energy of trimers estimated at 0.25 eV. The real breakthrough in determining long-range interactions between adatoms on a surface was made by Watanabe [19], who adopted the technique of first mapping out the

705

10.2 More recent studies

Number of observations

30 (a)

25

T = 400 K Non-interacting Experiments

20 15 10 5 0 0

10

(b1)

Fig. 10.7

20 30 40 50 Adatom separation (Å)

(b2)

60

70

(b3)

Re atoms on W(110) plane observed in FIM [18]. (a) Distribution of distances between two Re adatoms on W(110) plane after diffusion at 400 K. (b) FIM observations of Re3 trimer formation on W(110). (b1) Three Re atoms have been deposited on (110) plane. (b2) After heating at 420 K for 100 sec. (b3) After one more heating interval at 420 K, trimer has formed.

binding sites on the surface, and then measured the full two-dimensional distribution of separations with a reasonable number of observations, larger than 1000, to reduce scatter. In the first studies, a strongly bound atom was deposited at the center of the (110) plane of tungsten; a second mobile atom was then added [20]. This procedure lowers the total number of observations needed, as atoms are less likely to explore the plane edges. Additionally, if there are no interactions other than between nearest-neighbors the probability of occupation of any site other than next to the central one should be equal. Studied were the pairs Re-Pd as well as W-Pd. The distribution of palladium around a rhenium atom located at the center of the (110) plane is shown in Fig. 10.8a, obtained in 1638 observations after equilibration at 205 K. Here measurements over the entire surface have been replotted in the first quadrant. At this temperature, the root-mean-square displacement of palladium is comparable to the radius of the plane. Striking here are the long-range interactions and the lack of atoms at adjacent sites in the [100] and [1
10]
directions; dimers are orientated along the [111] direction. The free energy of interaction derived with Eqs. (10.9) and (10.10) from the distribution of atom separations, given in Fig. 10.8b, displays some remarkable features: along the direction, interactions are attractive out to a separation of 10 Å, interactions along and tend to be repulsive. Repulsive interactions are estimated at ~ 45 meV,

706

Atomic pair interactions

– [110]

>45 ~

40 200

8 – [110]

8

150

20 6

100

a /2

4 2

50

– [111] Re 2

(a)

4 2

– [111]

Re 2 4 –20

4

a /2

6 Free Energy, F (meV)

Number of Observations

250

6 8

6 8

10 T = 205 K a = 3.17 Å

–40 10

T = 205 K a = 3.17 Å

√2a /2

12

(b)

√2a /2

12 [001]

[001]

(c)

Free Energy, F(IRI) (meV)

40

Re-Pd

20

0

–20

–40 2 4 6 8 10 12 Interatomic distance IRI (a /2)

Fig. 10.8

FIM exploration of interactions between Pd and Re atom on W(110) [19]. (a) Distribution of Pd around stationary Re atom after diffusion at 205 K. (b) Interaction between Re and Pd adatom at different sites on W(110) plane. (c) Free energy of interaction as a function of the separation |R| [20].

while attraction at the nearest-neighbor site along amounted to − 35 meV. The magnitude of the interactions diminished with distance non-monotonically, and at the third nearest-neighbor site along , attraction amounted to 12 meV. If we plot free energy as a function of the radial separation of adatoms, oscillations are observed in Fig. 10.8c, but these are caused mostly by the surface anisotropy. Much the same features were found in the interactions of a tungsten atom with palladium on W(110). The distribution of separations is shown in Fig. 10.9a, and the

707

10.2 More recent studies

355 W-Pd

– [110]

– [110]

8

150

a /2

100 4 50

2

– [111]

W 2

6 ~40

25

a /2

4

2

– [111]

W 2 –25

4 6

4 –50

6

(a)

8

~40

50

6

Free Energy, F (meV)

Number of Observations

200

8 T = 225 K 10 a = 3.17 Å 12

8 T = 225 K 10 a = 3.17 Å

√2a /2

(b)

[001] √2a /2

12

(c)

[001] 60 W-Pd

Free Energy, F(IRI) (meV)

40

20

0

–20

–40

–60

Fig. 10.9

2 4 6 8 10 Interatomic distance IRI (a /2)

12

Interactions of Pd atom with stationary W atom on W(110) examined in FIM [20]. (a) Count of Pd adatoms around W adatom on W(110) plane after diffusion at 225 K. (b) Interaction energies between W adatom and Pd atom on W(110). (c) Free energy of interaction for Pd-W analyzed as a function of the separation |R|.

free energies of interaction in Fig. 10.9b. Overall features are quite similar to what has been shown before – attraction along , repulsion along and . Interactions oscillate along the direction; at the nearest-neighbor site they are around − 0.050 eV, in the second nearest-neighbor site in this direction they are close to

708

Atomic pair interactions

background (− 0.0028 eV), then in the third place attraction increases to − 0.0175 eV but falls to − 0.0064 eV at the next site. In Fig. 10.9c there is shown a superposition of all directions to give the dependence on the radial separation, which shows oscillations caused by the anisotropy of the surface. Nearest-neighbor interactions now are significantly stronger with tungsten (~50 meV) than previously with rhenium. Some rationalization of what is seen here has been offered by Stoneham [21] and also Kappus [22], who have stressed that interaction of an atom with the surface can induce changes in the position of surface atoms nearby. Kappus has pointed out that if a single adatom causes repulsion between surface atoms along and attraction along , then interactions between two such adatoms will be anisotropic – attractive along , but repulsive along and . However, in his continuous treatment, there are no indications of the oscillations actually found. Observations were subsequently also made on pairs of like atoms, iridium and rhenium, but with a larger data base of more than 2000 points, shown in Fig. 10.10a [23]. The analysis differs a bit now since both atoms are mobile and the distribution of non-interacting atoms, shown in Fig. 10.10b, is not uniform. The free energy of interaction for Ir-Ir on W(110) is obtained from the ratio of the two distributions in Eq. (10.9) and is plotted in Fig. 10.10c. What is clear here is that it looks quite similar to what was obtained earlier for hetero-pairs, but is stronger. Along , there is first a strong nearest-neighbor attraction, amounting to − 0.086 ± 0.002 eV, followed by a small repulsion of 0.0047 eVat a distance two spacings apart; attraction starts again in the third space at − 0.0272 eV, grows to − 0.0348 eV in the fourth, then diminishes to − 0.0156 eV and reaches background (− 0.0072 eV) in the sixth space. The interactions are much longer ranged than for hetero-pairs, and extend out to 14 Å. Interactions along and are initially repulsive at close spacings, but then turn to weakly attractive around three spacings. The radial changes in free energy as a function of separation between pairs is presented in Fig. 10.10d. Significantly different from past work [24–26] is the realization, based on studies by Chambers [23,27], that two next to each other iridium atoms resolved in the FIM actually occupy adjacent nearest-neighbor sites, and are not separated by two spacings. The same procedures as for the Ir pair were followed for rhenium. However, it was shown previously that Re dimers are not created on the W(110) surface [13,18,28], so the pair distribution is spread out over longer distances and a larger number of observations is in order. The Re-Re distribution as well as the free energy of interaction are shown on W(110) in Fig. 10.11, determined in 3145 observations; repulsion is noticeable everywhere at small separations up to 6 Å. There is quite a low atom population at the (1,1) position, and a lack at the close separations along and directions. The interaction starts to be attractive for bigger separations, and the largest number of observations is found at the (4,4) position, at a distance of 11 Å. Along , repulsion amounted to 0.0215 eV at the nearest-neighbor position, is stronger, 0.0554 eV, at the next spacing, and then changes to weak attraction, − 0.0164 eV. Attraction is − 0.0314 eV at the fourth, and then gradually decreases out to 16 Å and beyond. Along the orthogonal axes, that is along and , strong repulsion at the first spacings is replaced by quite weak attractions further out. What is clear in Fig. 10.11c is that for 6 Å< |R| ≤15 Å interactions

709

10.2 More recent studies

– [110]

– 12 [110] 200 Ir-Ir

100

a /2

8

4 – [111]

Ir

Number of Observations

Number of Observations

300

8

40

a /2 4

20 – [111] 0

4 4 8 8

(a) T = 375 K a = 3.17 Å

12

(b)

12

√2a /2

√2a /2 16

16

[001] [001]

– [110] a /2

Ir-Ir

50

8

4

50

– [111]

Ir –50

4

–100

Free Energy, F(IRI) (meV)

Free Energy, F (meV)

100

Ir-Ir

0

–50

8 0 12

(c)

(d) 16

4

8

12

16

20

24

28

Interatomic distance IRI (a /2)

√2a /2 [001]

Fig. 10.10

Pair interactions for iridium adatoms on W(110) surface, derived from distribution of interatomic distances measured in FIM [23]. (a) Distribution of distances between two Ir adatoms. (b) Distribution of distances between two identical non-interacting atoms. (c) Free energy of interaction between two Ir atoms after equilibration at 375 K. (d) Free energy of interaction for Ir2 as a function of the separation |R|.

710

Atomic pair interactions

– [110]

– [110] a /2

8

80 4

40

– [111] Re

a /2

8

80 4

40

– [111] Re

–40

4

4

8

(a)

12

Re-Re

Free Energy, F (meV)

Number of Observations

12 Re-Re

8

(b)

12

12 16

T = 390 K 16 a = 3.17 Å

√2a /2

20

√2a /2

20

[001]

[001]

Free Energy, F(IRI) (meV)

50 Re-Re 25

0

–25

(c) Fig. 10.11

0

4

8

12

16

20

24

28

32

36

Interatomic distance IRI (a /2)

Interactions between two Re adatoms on W(110) plane observed in FIM [23]. (a) Distribution of distances between two Re adatoms. (b) Free energy of interaction for Re2 dimers. Attraction is concentrated along [1
11]. (c) Free energy of interaction for Re2 as a function of the separation |R|.

are generally attractive but strongest along the close-packed direction. Effects somewhat similar to these findings have been arrived at in calculations by Dreyssé et al. [29]. However, in the experiments the transition from repulsive to attractive interactions occurs over a longer distance and the magnitude of the interactions is overestimated in the calculations. The type of interaction strongly depends on the plane examined: Re-Re pair interactions on the W(211) plane are attractive at close distances [30] with the atoms in adjacent channels or in the same one. This observation directly shows that interactions are surface mediated. Also studied were Ir and Re trimers, and for both it turned out that pair interactions could not account for the trimers observed at the surface. Based on pair interactions for Ir3, linear clusters should have a free energy of ~ − 0.170 eV, while the bent and triangular

711

10.2 More recent studies

form should be at ~ − 0.120 eV. From measurements of the lifetime for dissociation, trio interactions of − 0.130 ± 0.070 eV were deduced, indicating that many-body effects amounted to roughly a third of the trimer cohesion. Bigger many-body effects were found for Re trimers. Such trimers were observed stable at the surface, but taking into account only pair interactions, such trimers should have a repulsive free energy of 0.098 eV. However, from separate measurements it was found that the free energy of linear trimers amounted to − 0.240 eV [18] and around 0.030 eV lower for the bent form. Many-body interactions here amouted to − 0.340 eV for the linear and − 0.380 eV for the bent configurations. A few years later theoretical work on interactions between two tungsten adatoms on W(110) was done by Xu and Adams [31], using a fourth-moment approximation to tightbinding theory. The interaction energies for atoms placed at different sites are shown in Fig. 10.12. There are no detailed experimental studies of tungsten atoms with which to compare these predictions, but the trends in the calculations mirrored experiments on other adatoms. One worrying fact, however, is that two kinds of adsorption sites were seen at the surface, 0.11 Å apart. Two kinds of sites on this plane have never been observed in experiments. Additionally, it was claimed that the most stable adsorption place was three fold, while in experiments atoms sit only in four fold sites. Strong attractive interactions at near sites along are followed by small repulsions, and the same general patterns was also seen along and . However, the magnitude of the energies is too large. Nearest-neighbor energies have typically been measured less than − 0.10 eV, not − 2.6 eV found in this study. The strong angular anisotropy of interactions found in experiments on W(110) were, however, duplicated in these theoretical estimates.

[001] +0.003 +0.004 –0.005

–0.006

+0.011 +0.005

+0.003 +0.016

–0.075

? –2.633

+0.023 +0.056

+0.02 +0.025

+0.075 –0.02

Fig. 10.12

+0.008 –0.013

–0.004 –0.039

[111]

+0.05

+0.017

[110]

Pair interactions between two W adatoms on W(110) plane, calculated for different locations with a fourth-moment approximation to tight-binding (after Xu and Adams [31]).

712

Atomic pair interactions

E E F

D C F

Fig. 10.13

D A B C

F C

D E

B

A

F C

D E

D A B C

A: 85 meV B: 65 meV

E D C F

E F

C: –8 meV D: –28 meV E: 10 meV F: –11 meV

Effect of substitutional In atom (in black) on binding of nearby Cu adatoms, as calculated with EAM. Energies are relative to binding of Cu adatom on a clean Cu(111) surface (after Breeman et al. [32]).

Interesting but different calculations of the binding energy of atoms on Cu(111) were made by Breeman et al. [32] using an embedded atom method devised by Ackland et al. [33]. Their work was concerned with an In atom sitting in the first layer in place of a copper atom; it was found to have a strong effect on the energy of a Cu adatom nearby. As shown in the diagram in Fig. 10.13, the binding energy for a Cu adatom compared to that of a Cu adatom on a clean (111) plane was 0.085 eV higher. At other sites, however, the binding energy was significantly lowered, clearly revealing long-range interactions strongly dependent upon the orientation, and anticipating more elaborate calculations later on. Indirect estimates of pair interactions based on LEED and AES measurements were done by Kolaczkiewicz and Bauer [34] for Rh, Pt, and Ir on W(110). The values of the interactions were much higher than derived from direct measurements and have no oscillatory dependence on the distance. For Pt, nearest-neighbor interactions amounted to − 0.800 eV, while for Rh to − 0.600 eV. The second nearest-neighbor interactions were repulsive, amounting to 0.500 eV for Pt and 0.200 eV for Rh. Anisotropy of interactions was not taken into account in those studies, and direct measurements certainly provide more detailed insights. It is interesting to compare the work on Ir-Ir with more recent FIM studies of palladium interacting with another palladium atom on W(110), done with 1855 observations at T = 242 K by Koh [35,36]. The free energies of interaction are displayed in Fig. 10.14 and show the general trends already seen above. Along the close packed we first have strong attraction, − 0.0886 eV, followed by a repulsion of 0.0194 eV, and then for the next three spacings by significant attraction. Along and there is now general repulsion; this repulsion extends out significantly farther than for iridium. Interactions extend over a distance of 10 Å. However, as is clear from a comparison of the separation dependence of the free energy in Fig. 10.14c for iridium and palladium, behavior is very similar for the two. These similarities account for the linear chains seen with palladium clusters formed on W(110). What holds for all of the systems studied so far are the strong effects of orientation on interaction, with attraction along the orientation, and repulsion along the orthogonal axes. Pair interactions do not account for the structure observed for palladium clusters, since for bigger clusters the linear form has a significantly lower energy than do two-dimensional

713

10.2 More recent studies

30

562±19.8

25

Pd-Pd

– [110]

75

Pd-Pd

6 50

15

Free Energy (meV)

N(R)

20

– [110] 6

10 4 5

a /2

2 – [111]

0

2

25

– [111]

0 –25

2 4

–50

6 8

–75

2

a /2

4

√2a /2 10

–100

4

[001]

(b) 6

(a)

√2a /2

8 10 [001]

Separation (Å)

Free Energy (meV)

(c)

40

2.7

8.2

13.7

19.2

0

–40 Pd-Pd Ir-Ir –80

Fig. 10.14

Pair interactions for palladium on W(110) plane observed in FIM after equilibration at 242 K [35]. (a) Number of Pd pairs at different separations. (b) Free energy of interaction for Pd pairs at different locations. (c) Free energies of interaction for Pd-Pd and Ir-Ir as a function of atom separation.

islands. But 2D islands, not chains, were actually observed by FIM for clusters of more than eight atoms. The influence of three-body effects was investigated in detail. Trimers were observed in three configurations, linear, bent, and triangular, shown in Fig. 10.15a, so the free energy can be given as Flin ¼ 2Fð1; 1Þ þ Fð2; 2Þ þ F3lin

(10:15)

Ftri ¼ 2Fð1; 1Þ þ Fð2; 0Þ þ F3tri

(10:16)

Fbent ¼ 2Fð1; 1Þ þ Fð0; 2Þ þ F3bent ;

(10:17)

714

Atomic pair interactions

(a)

(b) ×2

×2 A ×2

B ×1 Bent

Linear ×2

L – [110]

×2

– [110]

Triangular

C ×4

[001]

–150 0

–100

×4

[001]

F3bent (meV)

(c)

D

–50

0

(d)

100

–100

F3tri (meV)

F3lin (meV)

Pd14 Pd on W(110)

–200

0

–100

Pd10

Pd8

Pd7

Pd4 –300 –150

–200 –300

–100

F3tri (meV) Fig. 10.15

–50 0

–200

–100

0

F3lin (meV)

Palladium clusters on W(110) plane [35]. (a) Schematics of trimer and (b) tetramer configurations. (c) Allowed values of F3tri, F3bent, and F3lin for Pd at 242 K. (d) Allowed values of F3lin and F3tri, based on the assumption F3tri = F3bent. Only needle-like white zone is allowed.

where Flin, Ftri, and Fbent denote the free energy of linear, triangular, and bent configurations respectively, F(x,y) the pair free energy for adatoms separated by the distance (x,y); F3lin, F3tri, F3bent give the three-body energy for linear, triangular and bent configurations. From experiment it is known that the trimer is always observed in a linear configuration. The probability Plin of observing the trimer in the linear configuration is given by Plin 4Ptri þ Pbent ;

(10:18)

where Ptri and Pbent denote the probabilities of observing triangular and bent configurations. The probability of observing a particular form depends exponentially on the free energy, Pbent ¼ CΩbent expð
Fbent =kTÞ;

(10:19)

715

10.2 More recent studies

where Ωbent is the number of energetically equivalent configurations with the bent shape, and C is given by normalization. Finally putting equations (10.15–19) together we arrive at Ωlin exp½
f2Fð1; 1Þ þ Fð2; 2Þ þ F3lin g=kT 4Ωtri exp½
f2Fð1; 1Þ þ Fð2; 0Þ þ F3tri g=kT

(10:20)

þΩbent exp½
f2Fð1; 1Þ þ Fð0; 2Þ þ Fbent g=kT: Information can therefore be deduced about the free energy of the different trimer forms. The same procedure was applied to tetramers in the forms A, B, C, D, and L in Fig. 10.15b, and also to higher clusters containing up to 14 atoms, giving us a set of restrictive equations. Combining all of them together restricted the allowed values of triofree energies, as shown in Fig. 10.15c for Pd on W(110). F3lin turned out to be attractive around − 0.060 eV or higher, a value comparable to the strongest pair interactions measured for this system (− 0.0886 eV). Assuming that the trio-energy of triangular F3tri and bent F3bent configurations is the same allowed values for F3tri and F3lin to fall into a narrow region shown in Fig. 10.15d. From this plot it is clear that the smallest trio contribution amounted to − 0.06 eV for F3lin and was around − 0.025 eV for F3tri. This study clearly showed the big influence of trio-interactions on cluster formation. The same procedure was applied to Ir clusters, where forming a linear chain is favorable up to 15 atoms. Working out all conditions, Koh again found a small strip for three-body interactions with the total free energy relatively constant, as shown in Fig. 10.16a–b. A comparison of the free energy of clusters obtained by taking into account only pair-interactions, and both pair- and trio-interactions, for Pd as well as Ir is shown in Fig. 10.16c-d. We can see that the fractional influence of trio-interactions increases with the size of the clusters. Trio interactions are attractive and of the same order as the strongest pair-interactions. Allowing even higher-order interactions probably would lower the strength estimated for trio-interactions. Many such effects have been measured on the bcc(110) plane. What happens on fcc(111) surfaces? The first STM investigation of pair interactions was done by Wahlström et al. [37] on Cu(111). The identification of atoms at the surface was tricky, as atoms emerged from the bulk during annealing at T > 800K. Comparison with other sources allowed the authors to suspect that they were dealing with sulphur atoms. Figure 10.17 shows the distribution of distances observed between atoms. For the pair interactions in Fig. 10.17b, at distances 21 ± 3 Å, 36 ± 4 Å and 66 ± 7 Å, three clear peaks were observed in 41 distances between 58 different atoms. The interaction energy was described by DEint ¼ C1

cosð2kF jRj þ ’Þ 3

jR j

þ C2

cosð2kF jRj þ ’Þ jRj2

;

(10:21)

where kF is the Fermi wave number and jRjthe pair separation. For this system, it meant a period of 15 Å at short distances and 30 Å at longer distances, where the second term dominated. The data are based on quite limited statistics and the segregation of sulphur at higher temperatures suggests the presence of sulphur in the sublayer, which would influence the measured pair interactions. That is why the quantitative data have to be

716

Atomic pair interactions

F3bent (meV)

(a)

–100

–150 0

–50

0

100

(b)

Ir15 0

–100

F3tri (meV)

F3lin (meV)

Ir on W(110)

–200

Ir14

–100

Ir16 Ir8

–300 –150 –100

–200 –300

–50

F3tri (meV)

–200

–100

0

0

F3lin (meV)

(c)

(d)

0

1000

Free Energy (meV)

Free Energy (meV)

Pair

–1000

Pair + Trio –2000

Pair

–1000

–3000

Pair + Trio

Pd

Ir –5000

–3000 2

4

6

8

10

12

Atoms in Cluster Fig. 10.16

14

2

4

6

8

10

12

14

16

Atoms in Cluster

Free energies of Ir clusters on W(110) derived from observations in FIM [35]. (a) Allowed values of F3tri, F3bent, and F3lin for Ir on W(110). (b) Allowed values of F3lin and F3tri for Ir shown in white. Estimates based on the assumption that F3tri = F3bent. (c)–(d) Contribution of pair- and triointeractions to cohesion of clusters on W(110). (c) For Pd, evaluated for (F3lin, F3tri) = (− 0.080 eV, − 0.034 eV). (d) For Ir clusters, evaluated with (F3lin, F3tri) = (− 0.180 eV, − 0.068 eV).

treated carefully. However this study is worth noticing as it is the first STM measurement of pair-interactions. In 1999, Levanov et al. [38] carried out RGL calculations of interactions between two atoms self-adsorbed on the (100) surface of fcc metals. The surface was modeled with 8 atomic layers consisting of 128 atoms. They evaluated the binding energy E1 for the atoms at the nearest-neighbor separation, and also at the next-nearest neighbor distance, E2, as well as E3 for atoms at third-neighbor positions. The results are given in Table 10.3,

717

10.2 More recent studies

Table 10.3 Binding energies (eV) between two adatoms self-adsorbed at first-, second-, and third-neighbor positions, obtained in RGL calculations on (100) plane [38]. Atom

E1

E2

E3

Cu Ni Ag Pd Pt Au

− 0.16 − 0.15 − 0.12 − 0.14 − 0.17 − 0.09

− 0.02 − 0.07 − 0.01 0.005 0.02 0.01

0.01 − 0.02 0.001 0.002 0.003 0.004

(a)

(b) 7

10

6 Count

8

5

6

4 3

4

2 2

1

0 0 Fig. 10.17

20

40

60

0 80 20 40 Distance between adatoms (Å)

60

80

Distribution of interatomic distances for adatoms on Cu(111) seen in STM experiments. (a) All measurements. (b) Distribution for atom pairs only (after Wahlström et al. [37]).

and indicate a fairly rapid decay except on Ni(100), where second range interactions (− 0.07 eV) are much stronger than for the rest of the materials investigated. The authors associated the behavior of Ni with stronger d-d interactions due to the unfilled d-shell. It is difficult, however, to judge this study since so far these are the only data for pair interactions on a fcc(100) surface. Estimates of interactions between silver adatoms on Ag(111) have been made in density-functional calculations by Fichthorn and Scheffler [39]. Work was done on a Ag(111) plane strained so the lattice constant was 4.61% smaller than for bulk silver, as well as on an ordinary Ag(111) surface. Interactions were modeled with a 4 × 4 × 4 slab, and at a fixed distance were assumed independent of the type of adsorption site (fcc or hcp) occupied by the atom. The fcc site was actually favored by 3 meV. Pair- and triointeractions were considered, with pairs up to 13th neighbor and trios up to 5th. Elastic interactions were found to be small, electronic effects playing the major role. The pairinteractions are plotted in Figs. 2.28 and 10.18, and clearly indicate strong attraction at nearest neighbors but also repulsive interactions ~3 spacings out. The repulsions on the strained surface were bigger than on an ordinary Ag(111), but the diffusion barriers were

718

Atomic pair interactions

Energy, meV 50 40 30 20 10 0 –50 –100 –200 ED = 52 meV

Fig. 10.18

Dependence of pair interaction energy on the separation from the central adatom for Ag on strained Ag(111) surface. Results obtained from density-functional theory calculations (after Fichthorn and Scheffler [39]). See color plate section for color version of this figure. E (meV) 150 75 0

Al Cu

E (meV) 75 50 25

Al

0

Cu 1

Fig. 10.19

2

3

4

5

6

IRI

Pair interactions on Al(111) and Cu(111) derived from density-functional calculations. Interactions for Al pairs on Al(111) and Cu on Cu(111). Distance jRj is given in terms of lattice site separation. Top graph gives results for frozen configuration, bottom for entirely relaxed one (after Bogicevic et al. [40]).

smaller. These interactions were found very important in that they led to island densities an order of magnitude larger than predicted in nucleation theory. Density-functional calculations were also done by Bogicevic et al. [40] using the VASP [41–43] program to examine pair-interactions of aluminum on Al(111) with a supercell 14 × 4 × 6, and copper on Cu(111) with a 12 × 4 × 4 cell. Cu(111) has a surface band of electrons, whereas Al(111) has no surface band states occupied. Results are given in Fig. 10.19 for interactions along the direction at the saddle points for diffusion and at equilibrium binding sites, in both the frozen configuration (at the top of the graph) and the relaxed (at the bottom). Variations in interatomic potential are present even when elastic response is frozen; this indicates

719

10.2 More recent studies

(a)

2500

Count

2000 1500 1000 (C1)

500 0

0

10

20

50 30 40 Distance (Å)

60

70

(b) 0.8

5 Pair potential (meV)

0.6

4

0.4

(C2)

0.2

3

0.0 –0.2

2

–0.4 20

30

40

50

60

70

20

30 40 50 Distance (Å)

60

70

1 0

(C3) 0

Fig. 10.20

10

Interactions between Cu adatoms on Cu(111) plane. (a) Distribution of pair separations deduced from STM measurements. (b) Energies for a pair of Cu atoms at different separations. Inset gives results for larger separations. Dotted line is fit of Eq. (10.22). (c) STM images of Cu atoms on Cu(111) at different separations. (c1) 12.9 Å (c2) 20.7 Å (c3) 26.7 Å. Local density of states decreases if there is a potential energy minimum between adatoms (after Repp et al. [45]).

that interactions are primarily electronic. Indirect elastic effects should predominate at larger separations, and were found to strongly affect diffusion and aggregation. They also show the island density with and without long-range interactions for Cu on Cu (111); interactions increased the island density by a factor of five. It should be noted, however, that Polop et al. [44] later claimed the effects of long-range interactions were overestimated in density-functional theory calculations. That has been examined by Repp et al. [45] in careful low-temperature experiments, using the scanning tunneling microscope, operated from 9–21 K with copper atoms on Cu(111). They measured the diffusion characteristics of copper atoms and were therefore in a position to define proper conditions for equilibration. More than 65 000 distances were tabulated to give the results shown in Fig. 10.20a. A one-dimensional plot of the free energy of interaction is now sufficient. After a peak below a distance of 10 Å, the energy falls and then oscillates steadily out to 70 Å, with a maximum swing of ~1 meV between peaks and valleys. An estimate was also made of a total barrier of 55 ± 5 meV to

720

Atomic pair interactions

the formation of more closely spaced pairs. The change in the density of states is illustrated in Fig. 10.20c, showing dimers separated by 12.9 Å, 20.7 Å, and 26.7 Å; the density shows protrusions or depressions due to maximum or minimum free energies between the atoms. There is, however, also the possibility of inelastic Friedel oscillations contributing [46] and the analysis did not touch problems of possible anisotropy in the interactions. Hyldgaard and Persson [47] at the same time arrived at a theoretical description of adsorbate–adsorbate interactions mediated by a surface-state band, which gave an energy  2 2 sinðδF Þ sinð2qF jRj þ 2δF Þ EðRÞ ¼
εF ; (10:22) π ðqF jRjÞ2 here εF is the Fermi energy measured from the bottom of the band, δF the Fermi-level phase shift which characterizes the adsorbate-induced standing surface wave function as observed in the STM, and qF gives the in-surface Fermi wave vector. Just as in the theory, the energy oscillates with a period equal to λF =2, half the Fermi wave length, the envelope of the magnitude decays as 1=jRj2 , and the phase shift does not change with separation jRj. The adsorbate–adsorbate energy mediated by the surface-state band, calculated from Eq. (10.22) for Cu(111) assuming a Fermi level phase shift δF ¼
π=2, is presented in Fig. 10.21, We can clearly see oscillations. However, there were differences between theory and observations, which were mentioned as possibly due to the neglect of electrostatic contributions to the interactions. It was found that scattering of surface state electrons from the adsorbates into the bulk states reduced interactions. 400

ΔEint(R) (meV)

ΔEint(R) (meV)

ΔEint2δ

200

ΔEint

10

0

–10 15

30

IRI (Å)

45

0 δF = –π/2

Cu(111) 0

15

30

45

60

IRI (Å) Fig. 10.21

Adatom–adatom interactions mediated by surface-state band. Solid curve gives results from Eq. (10.22), broken curve shows numerical approximation. Long-ranged asymptotic variation is shown in insert (after Hyldgaard and Persson [47]).

721

10.2 More recent studies

0

1

2

3

4

5

E(R) (meV)

6

7

8

9

(a)

Count

100 80 60 40 20 0

random

2 0 λF /2

–2

Count

60 (b)

40 20

E(R) (meV)

0 2 0 λF /2

–2

120 90 60 30 0

E(R) (meV)

Count

(c)

2 0 λF /2

–2 0

Fig. 10.22

1

2

3

4 5 IRI (nm)

6

7

8

9

Distribution of distances between adatoms as well as pair interactions on Cu(111) and Ag(111) deduced from observations with STM. (a) Cu adatoms on Cu(111), coverage 1.4 × 10−3 ML, T = 15.6 K. (b) Co on Cu(111), coverage 2 × 10−3 ML, T = 10.2 K. (c) Co on Ag(111), coverage 4 × 10−4 ML, T = 18.5 K. Dashed line gives fit to Eq. (10.22) (after Knorr et al. [48]).

A more extensive and more detailed study was carried out shortly thereafter by Knorr et al. [48], who again used scanning tunneling microscopy to look at the interactions between copper atoms on Cu(111) at 15.6 K, Co on Cu(111) at 10.2 K, and Co on Ag(111) at 18.5 K. Their results for the distribution of separations as well as the interaction energies are shown in Fig. 10.22. The jump rate of copper atoms was measured first as a function of temperature, and a temperature was then picked low enough so that the mean rate of jumps was below the rate of image recording. Furthermore, even though only 1.4 × 10–3 monolayers of copper were deposited, the distance between two atoms was counted only if no third atom or impurity was nearby. As a consequence of these precautions, there was excellent agreement with formula (10.22), indicated by dashed lines in Fig. 10.22, from long distances down to the first minimum, with stronger oscillations than found by Repp et al. [45]. For the maximum in the repulsive interactions

722

Atomic pair interactions

between two copper atoms before they formed a copper dimer Knorr et al. estimated a value of ≥13 meV. The experiments with cobalt on Cu(111) were done to establish how the chemical identity of the adatom affected interactions. The results were found to agree with those for copper on Cu(111) within the limit of error. The authors claimed that interactions were insensitive to the chemical identity of the adsorbate, which disagrees with the observation in earlier measurements for Re, Ir, or Pd pairs on W(110) by FIM. Rhenium pair interactions definitely differed from interactions between Pd and Ir pairs. The differences might be connected with the lack of an angular dependence on fcc(111) surfaces. Experiments with Co on Ag(111) were carried out to show that interactions arose from electronic origins, as Ag(111) has a surface band structure different from Cu(111) and λF =2 is 38 Å. It is clear that interactions on Ag(111) are much less pronounced and spread out over much larger spacings, and the “scaling of the interaction period with λF of the surface state clearly establishes that the long-range interactions are mediated by the nearly free two-dimensional electron gas of the surface state and excludes mediation via elastic lattice deformation”. Evidently the understanding of longrange two atom interactions is in reasonable shape, in large measure based on a knowledge of the diffusion characteristics, and the ability to see individual adatoms. Equation (10.22) turned out to describe quite nicely the observations made by STM, but this equation does not take into account the anisotropy of pair-interactions seen by Watanabe [20]. Hyldgaard and Einstein [49–51] worked out an expression for trio-interactions with experimentally accessible parameters in the case when the leading role came from electrons which scatter at all three locations and traverse the perimeter dT = d12+d23+d31. Assuming that adsorbates couple to the same local environment and have identical phase shifts δF pffiffiffi! 16 2 sinðqF dT þ 3δF
3π=4Þ 3 γ123 DEtrio ðd12 d23 d31 ; δF Þ ’
εF sin ðδF Þ ; (10:23) 5=2 π ðqF dT Þ5=2 3=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi here γ123 ¼ dT d12 d23 d31 is a shape dependent dimensionless ratio. It turns out that trio-interactions have an oscillatory character with slightly weaker amplitude than pair
5=2 and bulk-interactions, and decay a bit faster, as ~ dT . The oscillatory dependence of trio-interactions is illustrated in Fig. 10.23. Further calculations of the interactions between adatoms at long ranges have been carried out by Stepanyuk et al. [52] using density-functional theory and multiple scattering according to the Korringa–Kohn–Rostoker Green’s function method [53– 55]. They also used the “force theorem” and self-consistent spin-polarized potentials for single atom calculations. In the fully relaxed geometry, the distance of a Co atom from the surface is reduced by 14% compared to the ideal Cu layer distance. The substrate mediated interactions at large distances were found to be unmodified by relaxation and the calculations yielded the wave length λF. The local density of states at the Fermi energy is plotted in Fig. 10.24a for a single cobalt atom on a Cu(111) surface, showing strong oscillations with a period around 15 Å. In Fig. 10.24c is given an STM image of two cobalt atoms ~ 60 Å apart, and in 10.24b the oscillating interactions, determined in STM experiments and from theory, are compared. Theory agreed nicely with experiment, but for distances < 8 Å there are discrepancies with the predictions of Hyldgaard and Persson

723

10.2 More recent studies

0.2

ΔEt ri ple /εF

0.0 –0.2

δF = π/2

–0.4

d1 2 = d2 3 = d31 = dT / 3

–0.6 0.01 ΔEt r io /εF

ΔEt r io /εF = (ΔEtr i p le – ΔEp air s )/εF

0.10

0.05

0.00

–0.01

6

8 qF dT /π

10

0.00

–0.05 0

Fig. 10.23

2

4

6 8 qF dT /π = dT /(λF /2)

10

12

Three-atom interactions mediated by surface-state band. Top: full interaction of three atoms. Bottom: Trio-interaction energy. Solid curves give asymptotic results, dashed curves show numerical calculations. Insert details long-range variations (after Hyldgaard and Einstein [49]).

[47], probably due to their disregarding bulk electronic states in the scattering of surface electrons. The small, long-range interactions shown here have an effect on growth. Cobalt adatoms were found to diffuse alongside a chain of copper atoms, and eventually incorporated into the end of the chain, not into the side, much as has been found in the experiments of Koh [56], described in detail in Section 6.6. STM observations of cerium adatom layers on an Ag(111) surface were made by Silly et al. [57], who were able to obtain a self-assembled ordered hexagonal array at 3.9 K, shown in Fig. 10.25a. An STM image of a single cerium adatom revealed the oscillating electron density. From measurements of the location of cerium atoms mobile at 4.8 K, as in Fig. 10.25b, and the disappearance of the superlattice at 10 K, they were able to derive the distribution in Fig. 10.25c, which suggests a diffusion barrier of 0.011 eV, assuming a frequency prefactor of 1012 sec−1. The interaction potential responsible for positioning the atoms in the array is given in Fig. 10.25d and shows a first minimum at 32 Å with a depth of 0.8 meV, in agreement with the first-neighbor atom position, determined by the scattering properties of the adatoms. The repulsive barrier is in the range 0.3 meV at an adatom distance of 55 Å, so that 4.8 K is high enough to overcome such repulsions. To understand the creation of the superstructure, two- as well as three-body interactions have

724

Atomic pair interactions

2

10

1

Interaction energy (meV)

LDOS (EF ) (arb. units)

× 2.3 × 10–3

× 4.8 × 10–2 × 6 × 10–1

0

(a)

–1 0

× 3 × 10

Experiment

6

Theory Fit (Ref [47])

4 2 0 –2 –4

(b)

–6 10

20

30

40

–1

8 × 5 × 10–1

× 5 × 10

0

5

–3

10

15

20

25

IRI (Å)

Distance (Å) Co/Cu(111)

(c)

Fig. 10.24

5 nm

Co adatom interactions on Cu(111). (a) Local density of states for Co adatom on Cu(111), calculated with density-functional and K-K-R method. (b) Co pair interactions as a function of interatomic separation, derived from distribution of distances. Solid curve gives fit to Eq. (10.22). (c) STM image of two Co adatoms, indicating a sharing of density of states oscillations (after Stepanyuk et al. [52]).

to be taken into account. Results are in agreement with the finding of Knorr et al. [48] for Co on Ag(111). Understanding these long-range magnetic interactions is important, and can open up new possibilities for applications. Two years later, Negulyaev et al. [58] performed MC simulations and found a superlattice stabilized by a pair potential well with a depth of 0.6 meV for a coverage between 1% and 1.7% at temperatures < 5 K. At 8 K they observed collapse of the superlattice by formation of dimers. For higher coverages, a local hexagonal order was observed. Interactions between two palladium adatoms on a W(211) surface have recently been determined in field ion microscopic observations by Fu et al. [59]. Measurements were made of the number of atom pairs at different separations, at temperatures above 200 K, at which palladium diffusion is rapid enough to ensure equilibrium. When the two adatoms were in the same channel, a dimer tended to form with a free energy of − 0.0331 ± 0.0007 eV; in the next-nearest site the free energy was 0.1102 eV higher. Rather more impressive results were obtained when the two palladium adatoms were in adjacent channels. Here an attractive minimum was followed by repulsion out to 10 Å, then a small minimum was observed, followed by a rise in repulsion out to ~23 Å, as indicated in Fig. 10.26. The interactions for atoms separated by one or two empty channels were small.

725

10.2 More recent studies

(a)

Displacement (Å) –10

Counts

600

–5

0

5

10

(c)

400 200 0

Potential (meV)

(b)

Fig. 10.25

1 (d)

1st

0 λF /2 –1

0

50 100 Separation (Å)

STM images of Ce adatoms on Ag(111) plane self-assembled in a hexagonal superlattice at (a) 3.9 K and (b) 4.8 K. Superlattice is frozen at 3.9 K; at 4.8 K adatoms are mobile around binding sites. (c)–(d) Ce adatoms on Ag(111) surface. (c) Distribution of Ce displacements at 4.8 K around superlattice sites. (d) Pair interaction between Ce adtoms on Ag(111). First-neighbor position at 32 Å is shown by dashed line (after Silly et al. [57]).

0 1

Pair separation X along channels 3 4 5 6 7 8

2

9

15 10

F(X ) (meV)

5 0 –5 –10 –15 –20 –25 4 Fig. 10.26

6

8

10

12

14 16 18 20 Distance (Å)

22

24

26

Interaction between two Pd adatoms in adjacent channels of W(211) plane, deduced from FIM observations. Dashed line gives data fit to jRj
n with n = 1 (after Fu et al. [59]).

726

Atomic pair interactions

200

160

Count

120 80

160

Count

Experiment Simulation

120 80 40

40

0

0 0 100

5

10

80

15

20

25

30

n=1 n=2

5

10

15

20

25

30

35

60

0

5

10

15

20

25

30

35

60 40

F(meV)

F(meV)

0

80

40

20 0

20 –20 0

0

5

10

15

20

–20

25

30

–40

Distance (Å)

Distance (Å)

Fig. 10.27

Distribution of pair separations and of interactions for W adatoms on Ir(111) plane, derived from observations in FIM. Left: Along [2
1
1] axis. Right: Along [
101] axis. Fits with jRj
2 and jRj
1 dependence shown by dotted and solid curves. Simulations show the distribution for non-interacting W adatoms (after Fu et al. [60]).

A more extensive study by Fu et al. [60] appeared later with results again for two atoms in the same channel and also in neighboring ones. Unfortunately, there was no indication at all of the temperature of the measurements. The free energy of atoms in nearest-neighbor sites in the same channel was now given as − 0.070 eV, and in the next nearest-neighbor site the energy was 0.120 eV higher. With the two atoms in adjacent channels, the oscillations in the free energy are quite clear and can be fitted by the relation ð2kF jRj þ 2δF Þ=ð2kF jRjÞn , with kF = 0.25 Å−1 and δF ¼ 1:4. What is interesting here is that the best fit is obtained with n equal to one instead of two as expected from Hyldgaard and Persson [47]. The same kind of behavior was found for interactions between tungsten atoms on the Ir(111) surface. Shown in Fig. 10.27 are results along the [2
1
1] axis, and
also along [101]; both seem to conform to a plot with n equal to one again. In the direction [2
1
1], the biggest attraction was observed at 10.88 Å; the biggest repulsion was 0.07 eV. There was no attraction at the nearest-neighbor distance. In the direction [
101], the biggest attraction was observed at the next-nearest-neighbor site, at a distance of 9.42 Å; this amounted to 0.090 eV. Again there was no attraction at the closest possible distance. Lack of attraction at nearest-neighbor distances was observed previously for Re pairs on a W(110) surface [18,23]. Density-functional calculations of the interaction energies between silver atoms on a strained Ag(111) surface were carried out by Luo and Fichthorn [61]; the lattice constant was compressed by 4.61% to explore the possibility of adjusting layer formation. Electronic pair interactions obtained are plotted in Fig. 10.28a. They reveal strong attractive effects at the nearest- and next-nearest-neighbor positions. This region is followed by a ring of sites at which interactions are repulsive; this peaks at atomic separations of ~5 Å, and is then followed by a ring of attractive sites, past which, at a separation of ~15 to 16 Å, there is again a repulsive region. Finally, beyond this distance, attraction again sets in once more. Elastic interactions were also calculated, and are listed

727

10.2 More recent studies

Interaction (meV) 50 – 65 35 – 50 20 – 35 5 – 20 –5 – 5 –20 – –5 –35 – –20 –55 – –35 –200 – –130

(a)

(b)

500.0 Trio-Interaction Pair-Interaction

0.2 0 –0.2 –0.4 –0.6

(c) 0

5

10 15 dT (Å)

20

25

30

Angularly Averaged DFT Results

300.0

3

200.0

1

2

100.0 0.0 –100.0 –200.0

–0.8

Theory

400.0 Interaction (meV)

Triple Interaction (eV)

0.4

Fig. 10.28

Interaction (meV) 80 – 400 20 – 35 5 – 20 –5 – 5 –20 – –5 –35 – –20 –55 – –35

0

5

10

15

20

25

dT (Å)

(d)

–300.0

Ag-Ag interactions on Ag(111) plane. (a) Interactions between a Ag adatom and a central Ag on a strained Ag(111) plane as a function of the interatomic separation. Results obtained by densityfunctional calculations. Lattice constant of silver was 4.16% compressed. (b) Energies from Eq. (10.22) found in four-layer slab in density-functional calculations (kF = 0.28 Å−1, εF ¼ 0:662eV, δF = 0.45π) for central Ag adatom interacting with a second Ag in different positions. (c) Interactions of three adatoms on Ag(111) in their dependence on trio-perimeter dT. Total pair-energy indicated by circles. The squares show trio effects. (d) Comparison of interactions from Eq. (10.22) with angularly averaged DFT interactions from (b) (after Luo and Fichthorn [61]). See color plate section for color version of this figure.

together with electronic contributions in Table 10.4. Elastic effects are small, and for the 14th neighbor are below the resolution of the calculations. What is surprising, however, is that the magnitude of the interactions found here on the strained surface are much larger than seen in experiments on the unstrained surface. The period of the interactions amounted to ~11 Å, much smaller than the ~38 Å seen in experiments on an unstrained surface [48], giving an indication of the influence of strain on the interaction character. Luo and Fichthorn also looked at trio-interactions; Fig. 10.28c shows these interactions as a function of the trio perimeter dT. Trio-interactions are repulsive at close distances, attractive in the region 15–18 Å, but beyond 13Å are negligible. The calculations show that in this system substrate mediated interactions are primarily pairwise. The energy derived from Eq. (10.22) is given in Fig. 10.28b. The main differences are in the region of first- and second-neighbor distances and are caused by the formation of chemical bonds, not accounted for in Eq. (10.22). DFT calculations also show the anisotropy energy,

728

Atomic pair interactions

Table 10.4 Pair interactions obtained in density-functional calculations for Ag atoms on Ag(111) plane [61]. Neighbor

Distance jRj (Å)

Electronic (eV)

Elastic (eV)

1 4 5 9 11 14

2.84 4.91 5.67 7.50 8.50 9.82

− 0.1963 0.0564 0.0253 − 0.0249 − 0.0077 − 0.0096

− 0.0052 0.0007 0.002 0.0006 0.0022 − 0.0015

ε1

Fig. 10.29

ε2

ε3

ε4

Configurations of adatom pairs on Cu(111) plane, with energy designations at the top (after Stasevich et al. [62]).

which is not observed using Eq. (10.22), but was found previously in experiments [23]. After angular averaging the DFT results, agreement with the theoretical equation at longer distances is excellent, as in Fig. 10.28d. Stasevich et al. [62] have recently carried out density-functional calculations using the VASP program [41–43] for the interactions between copper atoms on Cu(111) and Cu(100). The configurations for two copper atoms and their designations are shown in Fig. 10.29. The supercells used in the calculations were (14 × 3 × 2). For this cell fourthneighbor interactions were ignored and the cell was referred to as (3 × 2); when a (14 × 4 × 2) cell was used, fourth-neighbor interactions were included and it was referred to as (4 × 2). The results on the two planes are given in Table 10.5; only data for the relaxed surface are shown, as the behavior of constrained surfaces is very similar. The results for (3 × 2) and (4 × 2) surfaces do not differ, demonstrating the negligible influence of fourthneighbor interactions. Second-neighbor interactions are significantly stronger on the (100) plane, but thereafter these effects drop off sharply. On Cu(111), second nearest-neighbor interactions are negligible, but on Cu(100) they are in the range of 1/7 the nearest-neighbor interactions. Interactions at third- and fourth nearest-neighbor distances were found to be negligible on both surfaces. The results are in good accord with the estimates for Cu(100) obtained by Levanov et al. [38] using RGL potentials, but do not agree with the experimental data of Knorr et al. [48] where longer-range interactions on Cu(111) were detected. The trio-interactions of Stasevich et al. for a triangular configuration of atoms on Cu(111) amounted to ~1/3 of nearest-neighbor pair-interactions, (0.117–0.083) ± 0.023 eV. Linear and bent configuration had an even smaller interaction energy, − 0.022 ± 0.011 eV and − 0.011 ± 0.011 eV. For Cu(100) the trio interaction energy for the linear configuration was about − 0.014 ± 0.011 eV, while for the bent one 0.051 ± 0.011 eV.

729

10.2 More recent studies

Table 10.5 Interaction energies obtained by ab-initio calculations between copper atoms in the configurations in Fig. 10.29 [62]. Cu(100)

Cu(111)

Interactions energy (eV)

(3 × 2)

(4 × 2)

(3 × 2)

(4 × 2)

ε1 ε2 ε3 ε4

− 0.332 ± 0.016 − 0.047 ± 0.009 − 0.003 ± 0.009

− 0.335 ± 0.012 − 0.043 ± 0.006 0.013 ± 0.008 0.002 ± 0.004

− 0.314 ± 0.019 0.004 ± 0.012 0.005 ± 0.006

− 0.323 ± 0.011 0.001 ± 0.012 0.003 ± 0.003 − 0.001 ± 0.003

T1 T2 I1

I2

T4

T3

T5

I3

T7

T6 C1

T8

I2C1

I1C1 C2

T10

T9

T11

I1C2 I3C1

T12 T13

(a)

T14

(b)

0.06

0.15

I1C1 I2

I2C1 C2

0.00

I3 I1C2

–0.05

I3C1

–0.10 I1

–0.15

Total Electronic Elastic

–0.20 –0.25 2

Fig. 10.30

3

4

5 6 7 8 d12 (Separation, Å)

9

Trio-Interaction (eV)

Pair-Interaction (eV)

C1

0.05

(c)

T5

0.04

0.10

T8

0.02 0.00 T4

–0.04 T2

–0.08

Total Electronic Elastic

–0.12

(d)

T14

T6 T 7

T3

–0.06

–0.10

10

T11 T13 T10 T12

T1

–0.02

T9

10

12

14

16 18 20 dT (Separation, Å)

22

Interactions between Al adatoms on the Al(110) plane derived from density-functional calculations. (a) Designation of pair-interactions. (b) Trio-interactions. (c) Dependence of pair-interactions on separation. (d) Effect of trio perimeter on trio-interactions (after Tiwary and Fichthorn [63]).

Recently, Tiwary and Fichthorn [63] looked at pair- and trio-interactions for aluminum on an Al(110) surface with DFT calculations implemented in the Vienna ab-initio simulation package VASP, investigating interactions for the configurations shown in Fig. 10.30a–b. Their data are presented in Table 10.6, and illustrated in Fig. 10.30c for

730

Atomic pair interactions

Table 10.6 Contributions to pair- and trio-energies in Fig. 10.30 calculated using DFT for Al atoms on Al(110) [63]. Interaction energy (eV) Pair/Trio

d12/dT

Total

Electronic

Elastic

I1 C1 I1C1 I2 I2C1 C2 I1C2 I3 I3C1 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 T14

2.86 4.05 4.96 5.72 7.01 8.09 8.58 8.59 9.49 11.45 11.87 14.83 15.64 16.19 16.78 17.17 17.59 18.00 19.36 19.54 19.83 20.17 20.55

− 0.104 0.038 0.033 0.029 0.036 − 0.002 − 0.005 0.015 − 0.004 − 0.006 − 0.060 − 0.044 − 0.021 0.032 − 0.019 − 0.025 0.011 0.019 − 0.005 0.015 0.004 0.017 − 0.017

− 0.237 − 0.008 0.008 0.003 − 0.001 − 0.002 − 0.004 − 0.002 0.003 0.054 − 0.006 − 0.003 − 0.002 0.007 − 0.001 − 0.004 0.002 0.002 0.000 0.003 0.002 0.000 0.000

0.133 0.046 0.026 0.026 0.037 0.000 − 0.001 0.017 − 0.007 − 0.060 − 0.054 − 0.041 − 0.019 0.025 − 0.018 − 0.020 0.008 0.017 − 0.006 0.012 0.003 0.017 − 0.017

pair and 10.30d for trio-interactions. For in-channel pair-interactions they observed a strong attraction, − 0.104 eV, at a nearest-neighbor separation, followed by weak repulsion of 0.029 eV at the second nearest-neighbor distance. Nearest-neighbor interactions have a mostly electronic origin, while at longer ranges interactions are due to elastic distortions of the lattice and start to play a leading role. Pair-interactions extend out to 8 Å; most trio-interactions, except for collinear trios, are also elastic in origin. Triointeractions have an oscillatory dependence on distance and were found to be important in understanding homoepitaxy on the Al(110) surface.

10.3

Summary With knowledge of surface diffusivities in hand, it has been possible to establish conditions for achieving an atom distribution at equilibrium, so the ability to obtain experimental insights into interactions between adatoms has become mature. It is quite clear now from direct experimental observations and also theoretical calculations that interactions between adatoms are not very strong but are of long range. More studies to define interactions can be

References

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expected to raise understanding in this area, and investigations of many-body interactions are certainly needed. With more information about the interplay between adatoms available it should also be possible to come to a better understanding of how surface diffusion proceeds with a finite concentration of atoms present on the surface. One thing that has become quite evident is that interactions are not short ranged, are likely to be anisotropic and influenced by strain. In addition, it is clear that pair-interactions are not enough to explain the creation of nanostructures on the surface. Moreover, from what has been said here it should be clear how closely surface diffusion and interactions are related. What will be most important in the future are clever experiments to quantitatively establish the magnitude of many-body effects between adatoms.

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Appendix

In both field ion microscopy and scanning tunneling microscopy, the preparation of sharp tips is crucial for carrying out observations with atomic resolution. The demands on the tip in scanning tunneling microscopy are not too rigorous; it has to have a sharp region of some kind in order to probe a surface with high resolution. For this task, tips are available commercially. In field ion microscopy, the tip is the surface to be studied, and therefore has to have a well-formed overall shape. As already pointed out in Chapter 2, there are fairly standard methods for preparing and polishing wire samples to the proper shape. We will give a few examples of how this can be accomplished. With the exception of the description for rhodium, these were all worked out by Liu [1], and with some minor changes taken directly from his thesis, but it should be noted that other recommendations are available [2–4]. Preparation of samples for field ion microscopy “Three conditions have to be met for a good field emitter specimen: (a) The end of the specimen, that is the tip, has to be sharp enough so that strong electric fields can be attained at reasonable voltages; (b) The tapered section, or shank, has to be smooth and devoid of large defects or undercuts, so that the emitter can withstand field stresses; and (c) The whole shank and tip should be smooth and symmetric to avoid serious distortion of the image. We describe in some detail the techniques for emitter preparation, and compare different techniques following the above guidelines, for five metals: tungsten, molybdenum, rhenium, iridium, and platinum.” In addition more recent findings for rhodium are described as well. A.1

Tungsten “The easiest and most successful method for preparing tungsten emitters is the drop-off technique. In this method a small section of 0.005 in diameter wire is dipped in 2N NaOH solution, using a ring-shaped platinum or nickel wire as counter-electrode, as shown in Fig. A.1. Approximately 12 V dc is applied across the cell by an automatic shut-off circuit, shown in Fig. 2.11. The tungsten wire is attacked most rapidly at the air-liquid interface, forming a neck there, as shown in Fig. A.2. As polishing proceeds, the portion of the wire below the liquid level is thinned and finally pinched-off at the thinnest place. When this happens the polishing current suddenly drops, as the surface interacting with the solution is greatly reduced; at this moment the potential across the cell is switched off

736

Appendix

Fig. A.1

Apparatus for electrochemical shaping of samples for field ion microscopy [1]. (a) For tungsten and molybdenum samples; (b) (11
20)-oriented rhenium wire; (c) arrangement for rhenium emitters; (d) (0001)-oriented rhenium; (e) iridium and platinum specimens.

immediately by the shut-off circuit to stop the back-etching. The wire is then lifted out of the solution and rinsed with distilled water, followed by ethyl alcohol, to remove the residual polishing solution still on the surface. The emitter tips obtained following this procedure ordinarily have a diameter of a few hundred Ångstroms, and the general result is very satisfactory. A.2

Molybdenum Molybdenum is located immediately above tungsten in the periodic table; chemically it is quite similar to tungsten. The polishing solution for tungsten (2N NaOH) also polishes molybdenum, but at a much higher rate. When the polishing conditions for tungsten are applied to molybdenum, the large ion currents cause turbulence in the solution which breaks the wire before it is properly dropped off. Attempts to reduce the current by either reducing the potential across the cell or the concentration of the polishing solution for polycrystalline wire lead to uneven etching of the wire. There is an established method for preparing molybdenum emitters: immerse 1/8 in of 0.003 in diameter molybdenum wire in 3.57N KOH, using nickel as a counter-electrode.

Appendix

Fig. A.2

737

Schematics of tip etching for (a) tungsten; (b) zone refined molybdenum; (c) (11
20)-oriented rhenium [1].

Then apply 4.0 V ac across the cell and polish until the lower section of the immersed wire acquires a long, uniformly thin shape. The potential is then reduced to 1.0 Vac and is applied as intermittent, short pulses while the shape of the wire is monitored with a stereomicroscope. At a magnification of 70X, the end of the wire seems to vanish as the wire is attacked from both the side and bottom. The rate of disappearance of the wire is directly related to the thickness of the wire. By stopping at the highest rate at which the wire disappears from view, a good emitter is usually guaranteed. There is only one disadvantage to this method: for wires of 0.005 in diameter, the section initially immersed has to be 1/4 in long in order to achieve satisfactory results. When high purity zonerefined single crystal wire is used, which is available only in 0.005 in. diameter, this is objectionable from an economic point of view. For zone-refined wires the following procedure was developed. Immerse 1/32 in of wire in either 2N NaOH or 3.57 KOH solution, using nickel as counter-electrode. Initially the wire is thinned by applying 12 to 15 V dc across the cell. After the wire becomes uniformly thin, as shown in Fig. A.2b, the voltage is lowered to 10 V and the polishing is continued until the lower section of the wire is barely visible in the stereomicroscope at maximum power (70X). The potential is then switched to 1–1.5 V ac and the procedure is finished as described in the previous etching method (Fig. A.2b). It is

738

Appendix

typical of this method of etching that the emitter does not appear sharp under the optical microscope at 400X. However, about 50% of the emitters prepared this way form a good field ion image at 5–8 kV. This method applies to both 0.003 and 0.005 in diameter wires and is indifferent to previous heat treatments of the wire. A.3

Rhenium Rhenium can easily be polished electrochemically using dc potentials and concentrated HNO3[5]. However, nitric acid tends to attack the air-liquid interface rapidly and good specimens are difficult to obtain consistently. Rhenium does not react readily with acids other than nitric. Most electrochemical polishing solutions cited in the literature [6,7] are fairly slow and time consuming. Besides this, rhenium responds to electrochemical polishing quite differently depending upon its heat treatment, increasing further the difficulty in developing an efficient and versatile etching technique. To resolve these difficulties, we have developed two methods for etching rhenium. For unheated wires, which are usually (11
20) oriented, the polishing solution consists of freshly mixed 20% H2O, 40% concentrated HNO3, and 40% HF (48% wt. conc.). In the cell, an iridium counter-electrode in the shape of a small loop about 1/4 in in diameter is immersed in the above electrolyte, as shown in Fig. A.1b. A Petri dish is used as the cell container and is filled with electrolyte up to the level of the iridium loop. When a 32 V dc potential is applied across the cell, polishing is rapid and the attack fastest at the interface. Frequent moving of the wire up and down results in a long, thin neck as shown in Fig. A.2c. The potential is then dropped to ~10 V dc and the polishing becomes uniform along the whole immersed section. Excessive bubbling at the anode (rhenium) is normal at this stage and should not be a matter of concern. After the wire is thinned and appears barely visible under the stereomicroscope, the old solution is abandoned and the final drop-off starts. In this stage a different electrolyte and cell are used: the latter is a narrow necked (5/16 in diameter) flat bottom glass flask containing ~2 cc of the electrolyte; the electrolyte consists of 4 parts concentrated H3PO4, 1 part glycerin, 1 part ethyl alcohol, and 1 part 10% aqueous HF. A single iridium wire serves as a counter-electrode, as shown in Fig. A.1c. A dc potential of 2.2 to 2.4 V is applied through the circuit shown in Fig. 2.11. The total current should be restricted to below 100 μA. The position of the air-liquid interface is essential for the success of the drop-off method. The interface should be located just above the thinnest portion of the wire, as shown in Fig. A.2c. Ordinarily a macroscopic undercut is left at the position of the meniscus (Fig. A.2c), since the drop-off solution also attacks the interface preferentially. This does not affect the performance of the tip: we have never lost an emitter due to this. For (0001) oriented zone-refined wires the final drop-off method is the same as described above. The initial thinning solution is different, however; it is made up from freshly mixed 40% glycolic acid, 30% conc. HNO3, and 30% HF (10% aqueous solution). The counter electrode is still the 1/4 in iridium loop, but the cell now consists only of a thin layer of electrolyte suspended on the loop, and the rhenium wire. The rhenium wire pierces the electrolyte layer, a small section protruding below it, as shown in Fig. A.1d. Only the section in the solution is etched. The potential across the cell varies from 3 to 4 V dc, depending upon the thickness of the electrolyte layer. Maximum polishing

Appendix

Fig. A.3

739

Schematic drawing of tip etching for (a) (0001)-oriented rhenium; (b) iridium; (c) platinum [1].

effect is reached when the ion current is 25 to 30 mA. Polishing is then rapid and smooth; a highly polished surface and a long, narrow section suitable for final drop-off are produced in a couple of minutes. Changes of the wire during polishing are shown schematically in Fig. A.3a. If incorrect proportions of electrolyte constituents are used, maximum polishing conditions cannot be achieved. This ordinarily results in preferential etching along certain crystallographic directions and hence the creation of deep furrows on the surface. Such undesirable effects usually can be removed by adding some conc. HNO3 to the electrolyte. The field emitters prepared in this way are very sharp. A perfectly smooth surface and clear field ion image can usually be obtained below 5 kV, using He as imaging gas. A.4

Iridium Iridium is famous for its inertness toward virtually all chemicals, including aqua regia. Reasonably rapid etching of iridium can be accomplished in molten NaCl or KCl salts [5], but control is difficult and the set-up is inconvenient. Chromic acid and ammonium carbonate have been used sequentially to make iridium field emitters [8–10]. This technique is quite slow and the chance of success is rather poor. Quite a simple method was reported by Graham et al. [11]; this uses dilute HF to achieve a thin, narrow taper and

740

Appendix

4N HCl to finally sharpen the emitter. The results are generally satisfactory, except that the thinning solution has no effect on heat treated iridium wires. To etch zone-refined wires we have developed a technique which yields excellent specimens with a high chance of success. This technique employs three different etchants. For convenience, the following solution can be made in advance and stored for indefinitely long periods of time: deposit iridium complex in 4% HF by electrochemically dissolving an iridium wire at 10 V dc and 2–3 A for about 1 hour, until the solution acquires a purplish-bluish color. The initial polishing solution is made of 1 part of the above solution, 1 part distilled water, and 2 parts of concentrated HNO3. The cell consists of a 1/4 in diameter tungsten loop as counter-electrode, immersed slightly below the electrolyte, as shown in Fig. A.1e. A section 1/16 in in length of 0.005 in diameter iridium wire is inserted into the electrolyte, and a potential ~ 40 V ac is applied across the cell. There should be appreciable noise due to sparking. Raise the voltage if this is not heard. The polishing is generally smooth and relatively fast compared to other methods. After 7 or 8 minutes a slender neck is formed below the interface, as shown in Fig. A.3b. The wire is then raised and the excess wire below the neck etched off. (If this stage is allowed to proceed, etching tends to localize near the meniscus.) The second step is to continue thinning in saturated (NH3)2CO3 solution. This is done simply by replacing the polishing solution and applying 2 V ac for two or three minutes, until the wire is barely visible under 70X magnification. The last stage is to drop off the lower section. This is done with an electrolyte which has been developed for preparing samples for transmission electron microscopy [12]: 1 part conc. H3PO4, 1 part conc. HNO3, and 1 part conc. H2SO4. Start out at 2 V ac and gradually decrease to 0.75 V ac as the wire gets thinner. The etching rate of this solution is fairly slow; approximately one hour is needed for the lower section to drop off. A bluish iridium complex tends to form near the anode in this final stage, blocking the view of the iridium wire. This complex does not affect the performance of the emitter, and can usually be removed by blowing air lightly on the solution surface. The above etching procedures are lengthy.1 However, the rewards are considerable: if the procedures are followed correctly a good specimen is guaranteed. Also, the sharpness of the tip can be controlled by controlling the size of the portion dropped off. Usually a good helium field ion image can be obtained at voltages ranging from 3 to 25 kV. 20% KCN can also be used for the final drop-off method. This saves considerable time since KCN attacks iridium at a much faster rate than the solution we used. However, KCN tends to leave a rough instead of a finely polished surface. Undercutting of the shank and uneven etching are more frequent and a good specimen is not always achieved. The chance of success is about 30% with KCN. A.5

Platinum Platinum is similar to iridium chemically except that it is less resistant to acids. Procedures for etching iridium can be applied to platinum directly. However, we have 1

The time required for the final stage has recently been reduced to ~10 min by applying a continuous ac potential of 0.75 V [13].

Appendix

741

discovered a much better and faster method to replace the lengthy procedures used with iridium. The construction of the electrolytic cell is similar to iridium, except that now an iridium counter-electrode is used. The electrolyte is replaced by a solution composed of equal volumes of water and saturated NH4Cl. The surface is highly polished and the etching is smooth and rapid when a potential of 30–40 V ac is applied across the cell. A thin slender neck is, as shown in Fig. A.3c, formed within one or two minutes. The potential is then dropped to 20 V ac and excess wire at the end etched off. The wire is further etched in the same solution at 2 V ac until the thin section becomes barely visible at 70X. Next the wire is taken out and immersed in distilled water to rinse off the ammonium chloride. The final drop-off procedure is the same as with iridium, but since platinum is less resistant than iridium, the time required is only 15 to 30 minutes. Field emitters prepared by this method ordinarily produce excellent helium ion images between 4 to 10 kV. The mechanical strength of platinum is much less than of tungsten or iridium. In order for the platinum specimen to withstand the stress of the applied field, which corresponds to a negative pressure of 105 atmospheres [5], it has to be very smooth and free of mechanical defects. The above procedures yield consistently successful specimens. 20% KCN can also be used for preparing platinum tips. However, the surface finish is far inferior to our method. Due to the roughness of the surface, less than 30% of the field emitters prepared by KCN survive the rigors of field evaporation even when exercising great care.” We should add that iridium etching with three different solutions can end up as quite a long process. For a quick procedure, producing good samples, a dip of the tip into a melt of 3 parts NaNO3 and 7 parts KOH yields good results very quickly. A.6

Rhodium Subsequent to Liu’s thesis, additional work has been done with rhodium, a material very similar to iridium and platinum. We have found differences in shape for tips etched from single crystal oriented wire and a polycrystalline sample. Both etch rapidly in a solution of 20% KCN and 12.5% NaOH, at a polishing potential of 20 Vac, which is used for creating a neck. For final preparation of the tip, the voltage is lowered to 2 V ac until drop-off occurs. This procedure is good enough to produce really sharp tips which will be imaged in the field ion microscope in a voltage range around ~5 kV. The shape of the tip obtained from polycrystalline and single crystal wire differs, however; the polycrystalline tip is longer. The tips obtained using this method are quite reproducible but it is not clear, at this moment, how mechanically strong they are. We have tried etching rhodium with a solution of molten NaNO3 and NaCl in the ratio of 4 to 1, and this solution is excellent for sharpening the tip in the final stages, but does not work well for creating the initial necking.

References [1] R. Liu, Chemisorption on perfect surfaces and structural defects, PhD Materials Science, University of Illinois at Urbana-Champaign, 1977. [2] A. J. Melmed, The art and science and other aspects of making sharp tips, J. Vac. Sci. Technol. B 9 (1991) 601–608.

742

Appendix

[3] A. J. Nam, A. Teren, T. A. Lusby, A. J. Melmed, Benign making of sharp tips for STM and FIM: Pt, Ir, Au, Pd, and Rh, J. Vac. Sci. Technol. B 13 (1995) 1556–1559. [4] F. Wu, P. Bellon, M. L. Lau, E. J. Lavernia, T. A. Lusby, A. J. Melmed, A new approach to preparing tips for atom probe field ion microscopy from powder material, Mater. Sci. Eng. A 327 (2002) 20–23. [5] E. W. Müller, T. T. Tsong, Field Ion Microscopy Principles and Applications (American Elsevier, New York, 1969). [6] A. J. Melmed, Helium field-ion microscopy of hexagonal close-packed metals, Surf. Sci. 8 (1967) 191–205. [7] W. Kollmar, D. Stark, Zur Feldemission der Metalle mit Hexagonaler Gitterstruktur, Z. Phys. 178 (1964) 39–43. [8] K. M. Bowkett, D. A. Smith, Field Ion Microscopy (North-Holland, Amsterdam, 1970) p. 57–61, Appendix 3. [9] M. A. Fortes, B. Ralph, The occurrence of glissile Shockley loops in field-ion specimens of iridium, Philos. Mag. 18 (1968) 787–805. [10] M. A. Fortes, B. Ralph, The growth of oxide on field-ion specimens of iridium, Proc. R. Soc. London A307 (1968) 431–448. [11] W. R. Graham, F. Hutchinson, J. J. Nadakavukaren, D. A. Reed, S. W. Schwenterly, Epitaxial deposition of platinum on iridium at low temperatures, J. Appl. Phys. 40 (1969) 3931–3936. [12] C. Cizek, F. Parizek, A. Orlova, J. Tousek, The homogeneization quenching of platinum, Czech. J. Phys. B20 (1970) 56–62. [13] J. Wrigley, Private Communication.

Index

ab initio, 446, 501 calculation, 79, 158, 264, 287, 289, 296, 403, 578 method, 578, 582 package, 646 simulation, 476 technique, 58, 578 VASP package, 364 VASP simulation, 326 abstractive chemisorption, 169 dissociation, 170 activated state, 18, 19, 20, 21, 56 activation energy 21 basic jump, 138 bridge jump, 268 cluster diffusion, 542, 559, 577, 581, 586, 589, 593, 609, 617, 618, 619, 635, 637, 644, 652, 665, 669, 671, 678, 687, 689 compact triangular to linear, 565 conversion, 138 corner rounding, 205 create misfit, 540 cross-channel jump, 192, 206, 229 descent, 440, 460, 477, 649 descent into kink, 441, 448, 473 detaching corner atom, 428 difference, 140 diffusion, 5, 31, 43, 49, 98, 100, 101, 102, 110, 111, 113, 119, 122, 127, 133, 138, 139, 183, 185, 187, 189, 191, 196, 201, 202, 205, 206, 209, 210, 211, 212, 213, 215, 216, 218, 219, 221, 223, 226, 227, 228, 230, 231, 233, 234, 235, 238, 264, 265, 266, 268, 269, 271, 272, 274, 275, 276, 277, 278, 279, 280, 281, 283, 284, 287, 289, 290, 291, 292, 295, 297, 298, 300, 302, 303, 304, 305, 308, 310, 311, 313, 314, 315, 317, 318, 319, 320, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 335, 336, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 350, 351, 352, 353, 354, 355, 357, 358, 359, 360, 361, 362, 365, 369, 370, 371, 402, 403, 425, 431, 444, 446, 449, 450, 452, 453, 457, 458, 462, 467, 472, 477, 498, 563, 575, 608, 634, 642, 650, 668, 675, 690

activation energy (cont.) diffusion with dislocation, 542 dimer diffusion, 561, 563, 571, 573, 578, 580, 581, 582, 595, 598, 599, 602, 608, 616, 632, 633, 634, 635, 639, 641, 643, 648 dimer exchange, 563 double jump, 142, 232, 236 double-exchange, 326 effective, 319, 606, 690 emergence, 199 escape, 431, 456, 458, 467, 468 exchange, 89, 90, 185, 192, 197, 206, 218, 226, 228, 236, 262, 264, 279, 286, 287, 294, 309, 323, 325, 327, 331, 334, 335, 363, 364, 367, 368, 371, 447, 455 exchange descent, 448, 465, 476, 477 for 18 atom cluster, 683 for 19 atom cluster, 684 for 61 atom cluster, 678 for 7 atom cluster, 683 gliding, 540, 596, 642 heptamer diffusion, 630, 643 hetero-dimer diffusion, 613 hexamer diffusion, 640 hopping, 197, 262, 263, 265, 278, 279, 280, 287, 288, 289, 290, 300, 305, 313, 317, 320, 324, 325, 354, 357, 358, 363, 364, 366, 367, 368, 369, 427 hopping descent, 443, 447, 449, 454, 455, 464, 476 in-channel ascent, 478 in-channel dimer diffusion, 617, 632 in-channel jump, 184, 185, 189, 192, 193, 194, 202, 204, 205, 210, 228, 232, 234, 456 in-channel motion, 88, 91, 186, 187, 190, 197, 199, 201, 204, 205, 206, 209, 215, 216, 218, 225, 227, 230, 233, 235, 238 incorporation, 432, 475, 580 local motion, 564 long jumps, 138, 139 long-range diffusion, 566 motion along step, 671 motion close to island, 498 motion cross-channel, 88, 94, 184, 187, 189, 194, 202, 224, 227

744

Index

activation energy (cont.) motion fcc-hcp, 320 motion hcp-fcc, 350 motion in-trough, 237 motion on sidewalls, 237 motion out of fcc, 350, 352 motion out of hcp, 352 motion perpendicular to step, 342 motion two-dimansional, 93 motion with misfit, 678 non-diagonal exchange, 272 pentamers diffusion, 595 rebound jump, 145 short-range movement, 620 single jump, 127, 141, 232, 236, 278, 335 spreading, 223 tetramer diffusion, 623 three-atom exchange, 263 trimer diffusion, 602, 607, 622, 640, 642 zigzag motion, 564 adatom catalyzed double exchange, 109 adatom density, 121 adatom island, 210, 442, 451, 671, 689 additional step-edge barrier, 431, 435, 439, 441, 442, 443, 448, 449, 450, 451, 452, 453, 455, 456, 458, 465, 467, 468, 476, 477, 478, 481, 485, 487 adsorbate-adsorbate mediated energy, 720 ad-trimer state, 108 AES, 323, 334, 368, 712 AFM. See Atomic Force Microscopy AFW approximation, 289, 309 parameters, 184, 190, 194, 195, 210, 235, 261, 278, 324, 357, 363, 366 Ag(001). See Ag(100) Ag(100), 104, 112, 150, 153, 155, 162, 169, 170, 322, 323, 324, 325, 326, 327, 328, 454, 455, 543, 547, 599, 600, 665, 667, 668, 671, 673, 675 unstrained, 112 Ag(110), 49, 98, 106, 119, 121, 207, 209, 210, 211, 455, 456, 601 (2 × 1), 601 (2 × 2), 210 Ag(111), 33, 48, 49, 51, 108, 110, 111, 162, 316, 317, 320, 321, 322, 446, 450, 451, 452, 453, 454, 455, 485, 575, 584, 598, 599, 601, 676, 677, 688, 689, 717, 721, 722, 723, 724, 726 strained, 717 unstrained, 51, 717 Al(100), 52, 103, 105, 261, 264, 265, 438, 543, 561, 562, 563 Al(110), 93, 184, 185, 438, 496, 729 Al(111), 52, 75, 164, 166, 168, 169, 265, 266, 267, 268, 269, 436, 438, 536, 563, 564, 566, 567, 679, 692, 718 argon ion scattering, 43 Arrhenius curve, 135

Arrhenius (cont.) plot, 100, 101, 105, 107, 110, 119, 127, 130, 133, 135, 138, 139, 142, 186, 191, 193, 195, 196, 199, 202, 205, 209, 219, 221, 227, 228, 230, 232, 233, 235, 236, 268, 275, 289, 290, 294, 297, 298, 299, 300, 302, 305, 308, 310, 311, 313, 318, 325, 329, 330, 332, 346, 350, 353, 358, 359, 360, 361, 362, 370, 371, 458, 465, 467, 480, 552, 559, 567, 575, 584, 589, 603, 604, 607, 608, 609, 621, 625, 643, 644, 645, 669, 672, 678, 690 Arrhenius dependence. See Arrhenius curve ARTwork computer program, 317 ascending lattice step, 456, 498, 506 step edge, 505 ascent mechanism atom exchange, 504 in-channel exchange, 649 asymmetric bridge site, 168 distribution, 15 motion, 13, 14, 15, 211 atom edge running, 596, 598, 664, 672, 675, 678, 691 atom pairs, 164, 167, 168, 169, 261, 724 Atom Probe. See Atom Probe microscopy Atom Probe microscopy, 27, 28, 29, 90, 96, 227 Atomic Force Microscopy, 34 atomic resolution, 34, 352, 469, 735 attempt frequency, 4, 87, 281, 290, 320, 321 angular, 321 attractive ring, 502 Au(100), 104, 112, 153, 367, 368, 425, 647, 648 (26 × 5), 425 hex, 365, 366 reconstructed, 647 Au(110), 107, 235, 236, 237, 478, 649, 651, 678 (1 × 2), 127, 158, 162, 232, 235, 236, 237, 649, 650, 692 Au(111),p 368, ffiffiffi 369, 370, 478, 590, 649, 676 (23  p3ffiffiffi), 114 (22  3), 370 reconstructed, 370, pffiffiffi 372 strained (23  3), 370 unreconstructed, 114, 370, 372 autocorrelation function, 30, 291 bakeout, 24, 35 ballistic energy, 109 transition, 121 barrier diffusion, 458, 539, 542, 561, 567, 569, 571 double jump, 288, 340 height, 20, 112, 113, 194, 197, 200, 205, 233, 235, 317, 324, 339, 364, 370, 372, 653, 654, 687 peak, 5 reflective, 41, 456, 458, 480, 482 single jump, 200, 236

Index

barrier (cont.) step-edge, 49, 430, 431, 432, 433, 435, 439, 440, 441, 442, 443, 448, 450, 451, 452, 453, 454, 455, 456, 457, 458, 465, 467, 469, 470, 477, 478, 479, 480, 481, 482, 483, 484, 485, 495 step-facet, 494 Be(0001), 426 Bi(111), 372 binding energy, 41, 49, 57, 78, 111, 113, 114, 115, 261, 351, 369, 423, 502, 617, 704, 712, 716 binomial theorem, 7, 8 Boltzmann term, 20 Boltzmann–Matano method, 201 bond breaking model, 669 Born–Mayer core repulsion, 57 bridge gliding, 633 bridge site energy, 304 Brownian particle, 118 bulk electronic states, 723 c(2 × 2) net, 103 cannon-ball mechanism, 168 cannon-ball-like path, 168 cannon-like trajectory, 166 canonical partition function, 697 capture number, 48, 53, 269 cartwheel glide, 542, 686 shuffle, 542, 686 CASTEP numerical code, 274 CEM. See corrective effective medium center of mass, 520, 521, 524, 525, 526, 529, 536, 547, 556, 561, 569, 573, 581, 582, 587, 590, 591, 594, 596, 602, 604, 606, 607, 625, 634, 635, 643, 690 channel plate, 25, 35, 36 chemical potential, 18 chemical surface reactions, 696 chemisorption, 164 classical partition function, 19 cluster atom-by-atom jump, 583, 599, 600 atom-by-atom motion, 583 compact, 536, 568, 599, 610, 630, 681, 682, 683, 685 hexagonal, 536, 538, 542, 573, 640, 682, 683, 685, 690, 691, 692, 693 large, 49, 517, 547, 550, 552, 553, 554, 664, 665, 668, 669, 671, 675, 676, 678, 679, 683, 692, 693 linear form, 590, 611, 653, 712 non-compact, 682 reshaping, 640, 673 rolling, 692 rotation, 686, 692 single jump, 654 stable, 49, 576, 676 translation, 686 cluster diffusivity prefactor, 553, 559, 568, 590, 617, 630, 672, 677, 683, 685, 686, 687, 692, 693

745

cluster frequency prefactor, 575, 578, 581, 589 cluster size, 84, 86, 441, 445, 473, 483, 499, 536, 547, 553, 569, 571, 578, 586, 596, 619, 637, 640, 644, 664, 670, 673, 687, 689 critical, 49, 188, 304, 309, 365, 370, 452 cluster–substrate coupling, 685 coalescence, 517 coarsening, 667 coefficient diffusion. See diffusivity friction, 119 scaling, 677 self-diffusion, 198 sticking, 443 cohesion, 27, 97 approximation, 272, 280, 288, 298, 310, 319, 326, 328, 331, 335, 336, 339, 340, 341, 348, 351, 354, 358, 364, 367, 370 energy, 303 cohesive energy, 57, 263, 272, 702 composition, 24, 27, 28, 29 compressibilities, 56 concentration, 2, 4, 31, 48, 49, 53, 54, 158, 169, 198, 201, 209, 223, 224, 291, 314, 327, 332, 333, 334, 336, 338, 342, 346, 347, 453 fluctuations, 30 gradient, 1, 4, 183 profile, 4, 53 concerted gliding, 587, 669, 687 hopping, 601 long-range motion, 566 motion, 478, 517, 536, 538, 562, 574, 584, 585, 586, 587, 588, 589, 592, 595, 599, 601, 633, 640, 646, 653, 692 sliding, 564, 583, 587 translation, 542, 583, 633, 640, 642 two-atom jumps, 599 configuration bent, 711, 714, 728 linear, 728 linear trimer, 565, 566, 569, 581, 584, 598, 600, 602, 623, 636, 639, 648 non-compact triangular trimer, 565 straight, 603 triangular, 714 triangular trimer, 565, 569, 576, 598, 601, 608, 623, 633, 635, 639, 640, 649 core breakup, 536, 550, 672 core-core pairwise repulsion, 57 corner breaking, 535 rounding, 205, 207, 550 corrected effective medium, 195, 206, 210, 286, 309, 325, 357 method, 57 theory, 118, 155, 276, 278, 297, 305, 313, 441, 444, 450, 453, 473, 475, 478 correlated motion, 107, 341, 404, 567, 578, 593, 654 coupling electronic states, 696

746

Index

covalent bonding, 104, 262 cross-channel ascent, 456, 478 barrier, 207 climb, 496 cluster, 528 descent, 456 diffusion, 185, 187 dimer, 525, 528, 533, 603, 607, 608, 630 direction, 439, 449 event, 90, 91, 210 form, 652 hopping, 647 mechanism, 230 motion, 89, 90, 91, 94, 186, 189, 194, 195, 204, 206, 210, 215, 216, 223, 230, 233, 234, 235, 236, 238, 455, 532, 632 movement, 90, 93, 94, 96, 98, 184, 187, 190, 194, 202, 206, 207, 211, 216, 230, 233, 237, 532, 630 process, 184 transition, 90, 94, 97, 98, 185, 209, 215, 496, 528, 532, 630 trimer, 527, 528, 607 crossing inner-corner, 496 outer-corner, 496 crowdion, 109, 110 crystal growth, 81, 156, 163, 431, 435, 442, 451, 467, 506, 599, 623, 696 Cu(100), 105, 107, 109, 119, 150, 155, 162, 272, 280, 283, 284, 285, 286, 287, 288, 290, 291, 293, 294, 295, 402, 429, 443, 447, 448, 455, 495, 496, 517, 543, 550, 575, 577, 578, 579, 582, 667, 670, 673, 675, 689, 728 Cu(110), 167, 194, 195, 197, 198, 200, 448, 449, 593, 692 (2 × 1), 592 Cu(111), 47, 48, 52, 78, 79, 80, 86, 109, 115, 121, 145, 155, 295, 296, 297, 298, 299, 300, 301, 402, 442, 443, 444, 445, 446, 487, 501, 502, 504, 575, 582, 584, 585, 586, 589, 590, 591, 673, 676, 688, 689, 690, 691, 712, 715, 718, 719, 720, 721, 722, 728 Cu(311), 194, 195 Cu(331), 194, 195 CY-EAM, 358 CY-XEAM1, 358 CY-XEAM2, 358 DACAPO code, 79, 296 dangling bonds, 169 de Broglie wavelength, 19 decacyclene, 145 hexa-tert-butyl, 145 decay rate, 451 defect, 46, 99, 155 concentration, 338 degeneracy, 683 density of islands, 49 density of states, 32, 57, 197, 504, 720, 722 density-functional

density-functional (cont.) calculation, 104, 112, 129, 210, 232, 263, 266, 268, 280, 309, 319, 334, 344, 358, 369, 426, 428, 449, 456, 473, 496, 504, 563, 580, 595, 600, 634, 639, 640, 679, 717, 718, 726, 728 estimates, 111, 234, 276, 280, 298, 361, 478 method, 288, 301, 307, 325, 357 theory, 56, 79, 112, 166, 185, 262, 264, 265, 268, 274, 289, 294, 298, 313, 319, 354, 357, 360, 364, 367, 368, 370, 436, 454, 504, 563, 564, 575, 679, 719, 722, 728, 729 descent multilayer step, 438, 487 prefactor, 431, 432, 465 descent barrier atom exchange, 443, 444, 445, 446, 447, 448, 449, 453, 454, 455, 456, 463, 464, 475, 476, 477, 478 kink exchange, 441, 463 vacancy exchange, 465 descent mechanism atom exchange, 194, 434, 435, 436, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 453, 454, 455, 458, 460, 465, 469, 472, 473, 475, 476, 477, 478, 487, 496, 504, 505 corner exchange, 447, 470 hopping over the edge, 194 in-channel exchange, 649 kink exchange, 441, 446, 447, 448, 449, 456, 470, 473 vacancy exchange, 465 desorption energy, 223 DFT. See density-functional theory diagonal motion, 584, 673 diffusion ad-dimer, 108, 364 cross-channel, 29, 89, 93, 98, 187, 190, 210, 216, 227, 230, 440 impact cascade, 162 in-channel, 91, 94, 98, 193, 195, 206, 207, 210, 218 inter-channel, 89 leapfrog, 529 long-range, 563, 564, 566, 569, 577, 582, 587, 594, 620, 623, 626, 644 non-local, 565 one-dimensional, 2, 44, 122, 123, 127, 225, 610 piecewise, 546 sequential atom jumps, 517 subsurface, 327 terrace limited, 517, 536, 667, 671 thermal, 146, 499, 575 transient, 146, 149, 151, 153, 155, 157, 158, 161, 162, 163, 164, 171 two-dimensional, 38, 44, 45, 91, 94, 117, 131, 184, 225 diffusion barrier atom exchange, 115, 191, 192, 194, 195, 197, 199, 200, 204, 205, 206, 210, 211, 234, 235, 236, 237, 238, 261, 262, 263, 264, 265, 278, 279, 280, 282, 285, 286, 287, 288, 289, 294, 295, 300, 307, 309, 323, 324, 325, 326, 327, 328, 331, 334, 335, 344, 354, 358, 364, 366, 368, 370, 449

Index

diffusion barrier (cont.) cross-channel, 630 dimer exchange, 563, 571, 578, 581, 595, 598, 600, 601, 632, 634, 637, 639, 648, 649, 651 hopping, 191, 195, 200, 206, 262, 263, 265, 272, 273, 276, 279, 280, 282, 285, 286, 287, 288 horizontal jump, 330 in-channel, 191, 195, 200, 207, 210, 230 tetramer exchange, 571 trimer exchange, 573, 593 vertical jump, 330 diffusion mechanism 180° dimer exchange, 542, 543, 637, 648 270° dimer exchange, 543, 637 360° dimer exchange, 543 along step exchange, 679, 692 atom exchange, 29, 51, 89, 90, 94, 96, 98, 99, 100, 101, 102, 103, 104, 105, 107, 108, 109, 112, 115, 117, 118, 162, 184, 185, 186, 189, 190, 191, 192, 193, 194, 195, 197, 198, 199, 200, 202, 205, 206, 207, 209, 211, 216, 218, 226, 227, 228, 229, 232, 233, 234, 235, 261, 262, 263, 264, 265, 268, 272, 276, 277, 278, 279, 280, 281, 283, 285, 286, 287, 288, 289, 290, 293, 297, 298, 299, 300, 301, 305, 307, 308, 309, 310, 311, 312, 314, 320, 321, 323, 324, 325, 326, 327, 328, 331, 334, 335, 341, 344, 353, 354, 358, 360, 361, 362, 363, 364, 365, 366, 367, 368, 370, 371, 372, 402, 403, 423, 425, 427, 428, 433, 456, 460, 478, 528, 531, 570, 571, 622, 634, 635 ballistic exchange, 162 between islands exchange, 49, 681 cluster exchange, 567, 570, 571, 576, 582, 593, 596, 600, 601, 653, 655, 669 concerted dimer exchange, 545, 630 concerted exchange, 272, 543 concerted hopping, 195 cooperative dimer exchange, 545 cooperative hopping, 545 correlated evaporation-condensation, 517, 536, 553, 669, 670, 673 cross-channel hopping, 190, 191, 195, 197, 229 detachment-attachment, 517, 549 diagonal exchange, 272 dimer exchange, 531, 532, 533, 534, 535, 542, 543, 544, 545, 561, 563, 564, 571, 573, 575, 578, 580, 581, 589, 593, 596, 598, 600, 622, 630, 632, 633, 634, 637, 639, 643, 647, 648, 652 dimer shearing, 517, 539, 550, 563, 566, 569, 570, 571, 577, 578, 579, 581, 588, 599, 600, 617, 623, 654, 669, 675, 687 double exchange, 107, 108, 288, 326 evaporation-condensation, 517, 536, 552, 664, 666, 667, 668, 671, 672, 673, 675, 676, 677, 678, 681, 685, 692, 693 gliding, 517, 586 hetero-dimer exchange, 620, 621 hopping, 103, 104, 109, 112, 115, 117, 121, 191, 192, 193, 194, 195, 197, 200, 202, 205, 211, 229, 235, 261, 262, 264, 265, 272, 275, 277, 278, 279, 280, 281, 282, 285, 286

747

diffusion mechanism (cont.) hopping rotation, 545 in-channel cluster exchange, 574, 593, 601, 646 in-channel hopping, 185, 186, 194, 197, 200, 202, 205, 210, 228, 232 internal-row shearing, 579 island exchange, 536, 548 jump-exchange, 106 jump-exchange-jump, 107 kink exchange, 679 leapfrog, 517, 574, 601, 637, 646, 649, 650, 654 long-range exchange, 265, 279, 325, 358, 360 multi-particle exchange, 263 multiple exchange, 105, 107, 109, 110, 493 non-diagonal exchange, 272 perimeter, 552, 668 quadruple exchange, 107, 288 rotation dimer exchange, 543, 545, 563, 580, 600, 637, 639, 647 shearing, 576, 577, 652, 653 sliding. See diffusion mechanism gliding sub-unit shearing, 687 tetramer exchange, 575, 582, 593, 598, 600, 601, 639, 651 three-atom exchange, 263 trimer exchange, 571, 575, 581, 589, 593, 596, 598, 600, 601, 635, 639, 643, 646, 647, 648, 649, 651, 652 trimer shearing, 517, 539, 550, 563, 570, 571, 577, 578, 600, 654, 675 triple exchange, 107, 288 two-atom dimer exchange, 639 vacancy-mediated exchange, 301 diffusivity, 1, 2, 4, 5, 7, 21, 22, 30, 41, 42, 46, 48, 49, 51, 53, 54, 86, 590, 596 cluster, 672 prefactor, 5, 43, 49, 58, 86, 88, 90, 91, 94, 98, 100, 101, 107, 122, 135, 139, 183, 184, 185, 186, 189, 190, 191, 192, 193, 194, 195, 196, 197, 199, 201, 202, 203, 204, 205, 206, 209, 210, 211, 212, 213, 215, 216, 218, 219, 220, 221, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 261, 262, 263, 264, 265, 266, 268, 269, 272, 274, 275, 276, 277, 278, 279, 280, 281, 283, 285, 287, 288, 289, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 312, 313, 314, 315, 317, 318, 320, 324, 325, 326, 328, 329, 330, 331, 332, 333, 335, 336, 338, 339, 340, 342, 344, 345, 346, 347, 348, 350, 352, 353, 354, 357, 358, 360, 361, 362, 363, 364, 365, 366, 369, 370, 371, 402, 403, 404, 425, 428, 432 dimer fcc-fcc, 79, 575, 587, 641, 645 fcc-hcp, 79, 575, 587 gliding, 578 hcp-hcp, 575, 645 mixed fcc-hcp, 564 rotation, 517, 531, 534, 610, 626, 633, 645 single jump, 602, 607, 608 translation, 531, 535, 564, 585, 610, 621, 626, 645

748

Index

dimer configuration staggered, 603, 604, 606, 617 straight, 604, 606, 617 dimer diffusivity prefactor, 602, 603, 604, 606, 607, 608, 616, 617, 620, 621, 627, 630, 646 dimer rotation, 535, 562, 564, 599 directional movement, 37 discommensuration line, 370, 372 dislocation, 28, 33, 506, 507, 509, 541, 542 core, 507, 508 mechanism, 304, 678 misfit, 678 movement, 517, 540, 573, 652, 678 nucleation, 539, 541, 692 displacement, 6, 9, 42, 43, 44, 45, 100, 103, 115, 122, 123, 133, 144, 156, 269, 304, 308, 321, 335, 372, 508, 524, 527, 531, 565, 606, 607, 611, 622, 627, 692 average, 13, 14, 165 ballistic, 162 Brownian, 676 center of mass, 520, 526, 531 cluster, 637, 682 concerted, 517, 570, 630, 686 fluctuation, 7, 9, 21, 41, 42, 43, 121, 522, 523, 524, 525, 526, 527, 528, 530, 531, 604, 607, 608 horizontal, 133 individual, 4 length, 6 mean, 521, 522 mean-square, 6, 15, 16, 41, 94, 100, 122, 223, 300, 305, 358, 360, 361, 371, 428, 507, 530, 531, 549, 567, 594, 607, 617, 625, 676, 677, 683 nearest-neighbor sites, 44, 682 overall, 5, 46 peripheral, 517, 535 random, 6 rate, 637 root-mean-square, 704, 705 sequentional, 568 square, 6 subsequent, 622 third moment, 13 variance, 522 vector, 5 dissipation, 118, 119 dissipation energy, 159, 162 dissociation, 162, 164, 165, 166, 167, 168, 169, 170, 171, 437, 533, 606, 607, 608, 610, 615, 617, 618, 619, 627, 650, 702 barrier, 650 cluster, 579, 685 dimer, 102, 649 dumbbell, 534 energy, 170, 569, 571, 606, 641 island, 51 lifetime, 711 rate, 48

dissociation (cont.) thermal, 51 dissociative adsorption, 169 dissolution, 696 distribution binomial, 10, 11, 15, 16 cluster displacements, 684 displacements, 7, 14, 43, 44, 45, 46, 94, 121, 122, 123, 125, 127, 133, 139, 141, 144, 232, 525, 531, 611, 622, 630 electron density, 56 Gaussian, 11, 15, 16 non-interacting atoms, 708 one-dimensional, 45 pair, 171, 701, 702, 708 random, 75, 148, 162, 163, 164, 166 separations, 699, 704, 706, 721 terrace length, 323 double jump frequency prefactor, 142 prefactor, 107 double rotation, 535 double shear glide, 640 double-hop, 107 double-shearing, 640 downward funneling, 51, 151, 155, 162 d-shell, 57, 402, 717 dumbbell, 89, 94, 97, 533, 534 dynamical misfit, 685 DYNAMO code, 121 EAM. See embedded atom method edge breakup, 548 diffusion, 596, 670, 671 effective barrier, 587 cluster movement, 690 descent, 478 diffusion, 291 dimer movement, 586 metastable walk, 236 tetramer movement, 588 trimer movement, 587 effective hopping integral, 57 effective medium approximation, 165, 639 theory, 57, 111, 195, 206, 235, 276, 279, 287, 309, 313, 315, 318, 325, 357, 359, 440, 443, 444, 447, 448, 449, 453, 454, 455, 475, 478, 492, 550, 561, 564, 579, 637, 670, 671, 673, 675, 686, 689 Ehrlich–Schwoebel barrier. See additional step-edge barrier Einstein relation, 7, 21, 121 elastic constant, 56, 57 effects, 696, 727 indirect effects, 719 lattice deformation, 722 theory, 509

749

Index

electron-charge distribution, 505 electro-polishing, 34 electrostatic attraction, 436 embedded atom method, 56, 58, 105, 109, 159, 190, 192, 197, 207, 228, 261, 265, 288, 297, 304, 309, 312, 317, 318, 325, 354, 358, 364, 440, 445, 446, 447, 448, 449, 454, 472, 473, 478, 517, 538, 541, 566, 578, 582, 600, 617, 639, 678, 686, 692 approximation, 57, 578 function, 57 simulation, 328 emission current, 30, 34 density, 29 empty zone, 41, 84, 85, 86, 157, 158, 467, 498, 499, 502 EMT. See effective medium theory enthalpy of sublimation, 403 enthalpy of vaporization, 403, 652 entropy, 5, 21, 75, 607 activation, 87, 122 equilibrium fluctuations, 553 evaporation-condensation, 672 FBD approximation, 275, 279 parameters, 275, 288 Fe(100), 269, 271, 272, 439, 440 Fe(110), 271, 272, 273, 439, 440, 567 Fe(111), 272 FEM. See field electron microscopy resolution, 30 Fermi energy, 720 level phase shift, 720 wave number, 715 wave vector, 720 wavelength, 720 Fick’s law first, 1 second, 2 field desorption, 102 field electron microscopy, 215, 217, 335, 338, 343 field emission current, 28, 342 fluctuation, 215, 216, 223, 225, 332, 333, 345 field emission microscopy, 53 field emission of electrons, 29 field evaporation, 27, 28, 36, 37, 39, 40, 41, 68, 98, 147, 187, 281, 353, 424, 425, 457, 507, 607, 630, 741 field ion micrograph, 147, 603 field ion microscope, 24, 29, 32, 34, 35, 36, 37, 41, 72, 86, 96, 123, 146, 148, 149, 152, 184, 232, 277, 281, 307, 336, 338, 356, 361, 432, 507, 623, 634, 741 field ion microscopy, ix, 27, 28, 33, 34, 43, 46, 48, 53, 64, 90, 98, 102, 123, 186, 187, 188, 192, 194, 202, 211, 212, 215, 217, 218, 219, 220, 221, 223, 225, 226, 227, 228, 230, 274, 277, 304, 305, 307, 328, 329, 335, 336, 338, 339, 340, 342, 345, 347, 348, 352, 354, 355, 402, 430, 434, 435, 456, 457, 458, 460, 467, 469, 504, 525, 527,

field ion microscopy (cont.) 537, 556, 593, 601, 602, 608, 610, 613, 614, 618, 634, 654, 664, 681, 698, 708, 712, 713, 722, 735 FIM. See field ion microscopy image, 39 magnification, 26, 39, 41, 698 resolution, 26, 64 Fireball, 58 Fokker–Planck formalism, 331 fourth moment, 13 Fowler–Nordheim relation, 29 free energy, 5, 81, 432, 433, 603, 604, 623, 676, 683, 706, 708, 710, 711, 712, 713, 714, 715, 724, 726 average, 683 change, 5, 21, 683 clusters, 715 Helmholtz, 697 interaction, 698, 705, 708, 719 pair, 714 Frenkel–Kontorova model, 541 frequency prefactor, 20, 21, 110, 111, 183, 196, 197, 207, 235, 266, 268, 269, 271, 278, 279, 281, 282, 284, 286, 288, 291, 293, 296, 297, 305, 308, 309, 311, 312, 314, 317, 318, 319, 320, 321, 322, 323, 324, 325, 327, 350, 352, 356, 359, 364, 365, 366, 368, 369, 370, 456, 458 frequency factor long jumps, 138, 139 single jumps, 127 frequency of occurrence, 683 frictional force, 127 Friedel oscillation, 698, 720 full-potential linear muffin-tin-orbital method, 318, 325, 351, 354, 366, 369 generalized gradient approximations. See GGA approximation getter, 35, 36 GGA approximation, 263, 264, 274, 276, 288, 289, 298, 301, 325, 357, 367, 454, 496, 564 ghost particle simulation, 116 gliding, 572, 573, 596, 598, 601, 630, 642, 648, 651, 652, 676, 678, 685, 686, 692 cluster row, 540 concerted row, 598 glue model, 57 half-chain process, 493 harmonic approximation, 320, 325 head-on-collision, 156 heat of condensation, 150 heat of sublimation, 56 helium atom scattering, 53, 54, 55, 98, 158 high resolution, 290 quasielastic, 194, 360 heptamer diffusivity prefactor, 637, 643

750

Index

heptamers gliding, 637 hetero-dimer, 530, 558, 613, 620 rotation, 612 hexagonal array, 723 core, 682 hollow four fold, 72 sites, 446, 561 three fold, 72 hopping between fcc sites, 276 in-channel, 226 short bridge, 272 hopping rate fcc-to-hcp, 296 hcp-to-fcc, 79, 296 horizontal dimer, 549 intermediate, 530, 531 states, 549 transition, 131, 132, 340, 530 “hot” atoms, 166 HRLEED. See LEED high resolution in-channel ascent, 478 barrier, 224, 226, 234 climb, 496 cluster, 575 diffusion, 210, 440 dimer, 525, 532, 593, 607, 608, 630 direction, 441 hopping, 197, 456, 575, 593, 596, 598, 632, 647 motion, 89, 187, 190, 191, 194, 195, 197, 202, 205, 206, 211, 218, 223, 227, 230, 233, 234, 235, 236, 237, 455, 456 movement, 91, 187, 193, 197, 211, 216, 226, 234, 236, 575, 630 tetramer, 528, 593 transition, 185, 229 trimer, 528, 529, 575, 607 incorporation, 433, 434, 435, 436, 439, 441, 443, 444, 445, 450, 455, 456, 458, 460, 461, 462, 463, 464, 472, 473, 475, 479, 480, 500, 502, 504, 506, 509 ascending step, 504 barrier, 461, 462, 476, 487, 505 descending step, 431 energetics, 436 lifetime, 481, 483 interactions adatom–adatom, 51, 267 adatom–step, 505 adsorbate–adsorbate mediated, 720 AFW, 233 attractive, 504, 698, 700, 701, 710, 711

interactions (cont.) bulk, 722 corrected effective medium, 325 cross-channel, 608 dipolar, 696 direct, 696 effective medium theory, 234, 317, 447 elastic, 436, 717, 726 electronic, 111, 696, 717, 726 embedded atom method, 156, 189, 193, 194, 195, 205, 235, 238, 288, 290, 298, 299, 317, 320, 325, 358, 360, 428, 441, 500, 543, 570, 571, 575, 585, 596, 600, 634, 639, 648 energy, 721 FBD, 190 indirect, 696 Lennard–Jones, 117, 192, 204, 262, 357, 363, 547, 595 long-range, 33, 48, 49, 52, 188, 205, 266, 300, 321, 502, 504, 593, 611, 653, 699, 702, 704, 705, 712, 719, 722, 723 long-range magnetic, 724 MD/MC-CEM, 293, 325, 327, 364 Morse, 363 pairwise, 189, 190, 703, 704, 727 quarto, 704 repulsive, 51, 425, 446, 504, 701, 704, 705, 717, 721 RGL, 109, 193, 195, 197, 200, 206, 210, 211, 228, 229, 234, 280, 282, 295, 300, 301, 320, 322, 326, 358, 360, 361, 368, 372, 449, 455, 462, 476, 544, 580, 590, 600, 630, 645, 648, 649 step-adatom, 51 step–step, 446 substrate–mediated, 696 tip-sample, 47 trio, 704, 722, 727, 728, 729, 730 van der Waals, 696 inter-cell concerted movement, 566 concerted sliding, 564 jump, 626 movement, 563, 582, 585, 599, 626, 631, 632, 641, 644, 645, 646 rotation, 631 inter-facets diffusion atom exchange, 492, 493, 494 chain exchange, 493 interior barrier, 440, 480, 481, 482, 483, 484, 485 interlayer atom exchange, 478 diffusion, 441, 444, 448, 451 jump, 478 mass transport, 443, 448 motion, 447, 454, 478, 496 movement, 49, 442, 443, 445, 449, 450, 476, 478, 487 process, 445 transport, 430, 442, 443, 476

751

Index

intermediate state, 262, 549 internal dimer rotation, 517, 579 internal three-atom shearing, 562 intra-cell concerted movement, 566 concerted rotation, 564 concerted sliding, 564 movement, 563, 564, 582, 631, 641, 645, 646 rotation, 564, 631 translation, 631 ionization potential, 27 Ir(100), 41, 100, 102, 103, 104, 353, 354, 458, 460, 619, 621, 622, 632, 633, 634, 653 Ir(110), 29, 90, 97, 195, 227, 228, 229, 634 Ir(111), 25, 38, 41, 72, 74, 86, 145, 152, 157, 272, 348, 350, 351, 352, 402, 434, 435, 436, 458, 460, 461, 462, 465, 467, 476, 478, 480, 487, 498, 499, 502, 542, 565, 567, 573, 577, 586, 589, 619, 627, 628, 630, 631, 633, 634, 640, 641, 652, 654, 681, 683, 686, 687, 691, 726 Ir(311), 227, 228 Ir(331), 228 island average separation, 49 coalescence, 121, 677 decay rate, 667 density, 49, 51, 52, 53, 121, 188, 205, 266, 267, 271, 278, 295, 302, 312, 316, 317, 320, 322, 355, 359, 365, 370, 452, 719 dissociation, 51 fluctuation, 49, 51, 429, 439 mean separation, 323 recombination, 51 rotation, 207 separation, 271, 284 jump basic, 137, 142 collective, 572 coordinated, 299 correlated, 121, 199, 272, 314, 319 cross-channel, 90, 94, 185, 192, 193, 195, 197, 204, 205, 206, 210, 211, 227, 229, 230, 234, 236, 237, 450, 598, 651 double, 12, 58, 107, 118, 119, 122, 123, 124, 127, 129, 131, 141, 142, 145, 187, 196, 210, 221, 223, 232, 236, 330, 501, 684 fcc-hcp-fcc, 299 horizontal, 140, 141, 330 in-channel, 185, 191, 193, 211, 233, 234, 236, 598, 601, 633, 651 inter-channel, 230 leapfrog, 593, 646 long, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 133, 135, 138, 139, 145, 210, 212, 215, 218, 236, 269, 319, 336, 339, 404, 630, 682, 684, 685, 686, 687, 692

jump (cont.) multiple, 200 nearest-neighbor. See single jump random, 64, 89, 147 rebound, 58, 121, 145, 215, 299 recrossing. See rebound jump single, 14, 16, 21, 22, 99, 107, 118, 121, 122, 123, 125, 126, 127, 129, 133, 135, 137, 138, 139, 141, 142, 145, 187, 199, 200, 209, 210, 215, 219, 269, 277, 330, 336, 339, 361, 626 triangular trimer, 589 triple, 126, 141, 220, 684 uncorrelated, 668 vertical, 131, 133, 140, 141, 330, 336 jump length, 1, 4, 5, 6, 87, 274, 685 characteristic, 121 mean, 116 mean-square, 6, 21, 122, 183 nearest-neighbor, 7, 12, 87, 122 jump rate, 4, 9, 13, 17, 20, 21, 22, 41, 42, 45, 123, 135, 139, 144, 196, 232, 311, 356, 504, 522, 524, 525, 526, 530, 550, 575, 584, 604, 607, 608, 721 transients, 44 kinetic Monte Carlo simulation, 207, 210, 267, 289, 300, 370, 581, 585, 679 kink breaking, 672 escape, 550 motion, 540, 678 KMC simulations. See kinetic Monte Carlo simulation Kolmogorov relation, 520 Korringa–Kohn–Rostoker Green’s function method, 722 Kronecker delta, 604, 699 large cluster power law, 664, 671, 678 lateral translation, 150 lattice strain, 111 layer-by-layer growth, 150, 151, 152, 443, 448 LDA approximation. See local-density-functional approximation leapfrog jump-up, 574 transition, 574, 646 LEED, 43, 64, 151, 309, 311, 334, 712 high resolution, 284, 323 spot profile, 295, 355, 372, 429 LEIS, 328 Lennard–Jones crystal, 89, 116, 117, 157 fcc(111) plane, 156 minimum, 560 system, 149

752

Index

lifetime, 17, 223, 479, 480, 481, 482, 483, 485, 649 activated state, 17 metastable configuration, 637 L-J potentials. See Lennard–Jones potentail local-density-functional approximation, 262, 264, 288, 289, 325, 357, 367, 454, 563 calculation, 359, 561 method, 355 theory, 185, 261 long jump prefactor, 135, 138, 139 long-distance transition, 299 long-range effect, 503 mechanism, 493 transition, 288 translation, 575 low-energy ion scattering, 211 map (1 × 1), 102, 308 adsorption places, 39, 41, 72, 103 binding sites, 72, 78, 623 sites occupied, 38, 427, 634 sites visited, 278 MC simulation. See Monte Carlo simulation MD simulation. See molecular dynamics simulation MD/MC-CEM, 57, 195, 206, 210, 279, 281, 282, 286, 293, 308, 309, 311, 319, 324, 366, 368, 590 mean free path, 26 mean-field theory, 48, 49, 668 mean-square separation, 699, 700 melting point, 89, 106, 116, 126, 141, 238, 262 simulations, 596 temperature, 121, 573, 582, 596, 642 metastable location, 72 walk, 47, 128, 129, 227, 236, 428, 529 Meyer–Neidel rule, 318 microchannel plate. See channel plate mixed hcp-fcc dimer, 587 Mo(100), 303, 322 Mo(110), 273, 302, 303 Mo(111), 303 modified Bessel function, 14, 15, 30, 122 molecular dynamics, 57, 106, 117, 195, 263, 265, 276, 277, 279, 288, 305, 306, 309, 312, 315, 317, 320, 325, 358, 403, 426, 453, 501, 572, 596, 600, 643, 647, 648, 649, 673, 685 action-driven, 109, 571, 575, 581, 589 calculation, 118, 364 classical, 358 estimates, 89, 200, 207, 237, 275, 364, 644 investigation, 331 Langevin, 293 results, 350 semi-empirical, 357

molecular dynamics (cont.) simulation, 89, 98, 105, 107, 115, 119, 121, 127, 142, 145, 149, 159, 162, 165, 185, 195, 197, 199, 200, 206, 211, 232, 234, 236, 237, 262, 272, 276, 279, 280, 282, 286, 287, 288, 289, 291, 293, 295, 297, 299, 300, 301, 305, 306, 310, 311, 313, 314, 318, 319, 320, 321, 322, 325, 326, 328, 331, 358, 360, 361, 365, 366, 368, 369, 370, 372, 465, 473, 542, 545, 559, 560, 563, 566, 578, 580, 584, 585, 586, 588, 589, 590, 593, 599, 600, 630, 631, 637, 639, 640, 642, 643, 644, 645, 646, 647, 648, 649, 686 study, 97, 108, 193, 199, 350, 354, 369, 600, 678 VASP package, 357 molecular statics, 197, 200, 289, 298, 438, 440, 449, 494, 501, 635 constrained, 108 moment generating function, 12, 15, 525 Monte Carlo bond-counting model, 446 calculation, 42, 550 estimates, 44, 475 modeling, 197 simulation, 41, 52, 57, 123, 147, 158, 159, 162, 235, 264, 268, 271, 278, 298, 300, 305, 309, 317, 320, 323, 355, 363, 365, 403, 429, 439, 443, 446, 451, 480, 501, 502, 553, 578, 579, 604, 666, 668, 669, 670, 671, 673, 675, 677, 678, 687, 689, 690, 724 TST, 218 Morse calculations, 194 function, 56, 217 movement atom-by-atom, 534, 535, 587, 588, 589, 599, 626, 692 concerted, 566, 585, 590, 596, 647 fcc-hcp, 572, 582, 588, 590, 596, 686 hcp-fcc, 587 hcp-hcp, 265, 299 hcp-hcp to fcc-hcp sites, 645 local, 564, 575, 617, 620, 633, 641 peripheral, 547, 573, 617, 627, 664, 665, 666, 668, 672, 673, 675, 678, 687, 689, 693 step edge, 589 vacancy, 552 multilayer barrier, 496 step, 495 transitions, 498 multilayer descent, 446, 495 atom exchange, 494, 496 nearest-neighbor transition, 15, 98, 119, 122, 127, 135, 138 NEB method. See nudged elastic band method negative step-edge barrier, 455, 487 net (1 × 1), 103 neutron scattering, 54 Ni(100), 49, 102, 103, 104, 105, 153, 278, 279, 280, 281, 282, 283, 428, 440, 441, 442, 543, 561, 569, 570, 571, 717 paramagnetic, 280

Index

Ni(110), 91, 155, 168, 186, 187, 188, 189, 190, 191, 192, 193, 425, 428, 440, 441, 442, 575, 601 Ni(111), 72, 274, 275, 276, 277, 279, 428, 440, 441, 442, 501, 540, 571, 572, 573 Ni(311), 91, 193, 194 Ni(331), 91 non-equivalent paths, 686 processes, 630, 686 non-uniform step-edge barrier, 451 nuclear resonance scattering, 54, 334 nucleation theory, 51, 53, 184, 266, 267, 269, 281, 284, 296, 304, 312, 315, 316, 317, 320, 359, 372, 452, 718 mean-field, 429 nudged elastic band method, 109, 264, 272, 300, 326, 358, 493, 690 one-dimensional arrangement, 632 chain, 502 cluster, 619, 646 cluster diffusion, 601 crystal, 146 island, 204, 541 lattice, 146 metastable chain, 502 model, 146 motion, 21, 520, 530, 685 movement, 89, 186, 193, 195, 228, 231, 235 plane, 39, 141, 480 process, 22 surface, 183, 592 system, 123 theory, 685 trimer diffusion, 526 Ostwald ripening, 667 PACS. See perturbed γ–γ angular correlation spectroscopy partition function, 18, 19, 20, 21, 698 Pb(110), 238, 478 Pb(111), 478 Pd(100), 104, 151, 153, 158, 309, 310, 311, 312, 598 Pd(110), 204, 205, 206, 207, 596, 598 Pd(111), 171, 312, 313, 314, 450, 596, 598 Pd(311), 205 Pd(331), 205 pentamer rotation, 626 translation, 626, 633 perimeter energy, 553 perturbed γ–γ angular correlation spectroscopy, 53, 54, 281, 290 phase shifts, 722 photoemission electron microscopy, 314

753

piecewise rotation, 546 plane-wave pseudo-potential, 325 potential Ackland. See potential ATVF AFW, 189, 193, 194, 195, 205, 233, 235, 278, 285, 297, 366, 369 ATVF, 190, 279, 344, 712 bond-order, 642 Ceperley–Alder, 563 corrected effective medium, 152, 286 EAM5, 191 effective medium theory, 191, 210, 236, 263, 265, 280, 285, 296, 297, 317, 321, 357, 366, 575, 647, 689 embedded atom method, 75, 79, 107, 115, 121, 159, 184, 190, 192, 197, 203, 205, 209, 210, 211, 228, 233, 234, 272, 275, 276, 279, 280, 286, 287, 288, 289, 297, 298, 299, 300, 310, 311, 317, 320, 324, 325, 326, 327, 331, 350, 357, 358, 363, 364, 366, 367, 369, 372, 403, 436, 444, 446, 447, 448, 453, 477, 496, 501, 539, 540, 542, 563, 566, 567, 569, 571, 572, 573, 578, 581, 582, 589, 592, 596, 599, 600, 601, 632, 637, 642, 643, 646, 647, 648, 690 Ercolessi–Adams glue, 263 FBD, 325, 454 Finnis-Sinclair, 444, 447 glue, 114, 370 hybrid tight-binding-like, 162 Lennard–Jones, 56, 89, 112, 117, 118, 185, 189, 193, 197, 203, 204, 206, 210, 218, 229, 233, 234, 236, 264, 265, 268, 275, 276, 278, 280, 289, 299, 300, 305, 306, 307, 310, 313, 320, 326, 331, 351, 352, 354, 357, 358, 363, 364, 368, 370, 559, 560, 664 many-body, 185, 189, 204, 205, 217, 218, 226, 229, 234, 235, 276, 287, 297, 305, 306, 319, 327 MD/MC–CEM, 191, 287, 309 Morse, 118, 119, 189, 190, 193, 194, 203, 204, 205, 209, 217, 218, 225, 226, 228, 230, 231, 233, 234, 235, 272, 275, 278, 279, 285, 287, 297, 300, 303, 305, 306, 309, 312, 317, 324, 328, 330, 331, 344, 345, 350, 354, 355, 357, 363, 366, 369, 546 Oh and Johnson. See potential OJ OJ, 190 one-dimensional, 17, 20, 118 pair, 56, 268, 306 RGL, 57, 106, 107, 127, 192, 197, 199, 204, 206, 207, 210, 211, 229, 232, 234, 236, 237, 277, 288, 289, 294, 297, 298, 300, 307, 310, 312, 313, 314, 315, 318, 319, 328, 351, 352, 354, 358, 364, 365, 367, 370, 448, 455, 463, 493, 501, 529, 582, 596, 630, 631, 646, 648, 649, 728 semi-empirical, 56, 98, 357, 403, 448 surface-embedded atom, 326, 543, 600, 639 Sutton-Chen, 275, 279, 306, 364, 537, 686, 687 VC, 57, 185, 189, 190, 193, 205, 233, 234, 235, 261, 285, 288, 309, 312, 325, 363, 366, 369, 454 precursor, 167, 170

754

Index

probability, 7, 9, 10, 74, 81, 96, 122, 467, 498, 520, 521, 522, 523, 524, 527, 531, 533, 604, 697, 698, 714 converting, 137 finding a displacement x, 15 incorporating, 498 long jump, 118, 119 occupation, 705 recondensation, 667 reconfiguration, 610 rotation, 610 Pt(100), 100, 101, 104, 107, 109, 115, 153, 215, 361, 362, 363, 364, 365, 427, 429, 475, 542, 543, 634, 637, 639 hex, 362, 363 Pt(110), 230, 231, 232, 233, 234, 235, 475, 647 (1 × 2), 127, 129, 232, 234, 428, 528, 529, 637, 646 Pt(111), 49, 78, 84, 86, 110, 111, 112, 148, 155, 156, 162, 163, 166, 167, 169, 317, 319, 355, 356, 357, 358, 359, 360, 361, 402, 426, 429, 450, 453, 457, 467, 468, 469, 472, 473, 475, 476, 477, 480, 485, 486, 487, 500, 639, 640, 641, 642, 643, 645, 646, 652, 654, 687 Pt(311), 232, 233, 234 Pt(331), 232, 233, 234 pump mechanical, 35 mercury diffusion, 35 quantum confinement of electrons, 502 quantum state, 697 quasi-harmonic approximation, 325 quenched molecular dynamics simulation, 210 random walk, 16, 121, 525, 547 rate basic jump, 137 coarsening, 668 core breakup, 548, 549 corner rounding, 548 cross-channel motion, 94 deposition, 469 dimer nucleation, 483 double jump, 135, 142 escape, 431, 456, 467, 468, 479, 480 evaporation, 669, 677 evaporation–condensation, 675 horizontal jump, 129, 135, 140 impingement, 49 in-channel motion, 94 incorporation, 48, 431, 460 kink escape, 548 long jump, 137 nearest-neighbor transition, 122 rotation, 626 shearing, 550

rate (cont.) single jump, 122, 135, 137, 138, 139, 141, 142 vertical jump, 135, 140 ratio charge-to-mass, 90 double-to-single, 123, 125, 126, 127, 223 Re(0001), 80, 263, 347, 348, 404, 457, 618, 619, 652 reaction coordinate, 17 rebound energy, 156 transition, 144 rebound jump frequency prefactor, 145 reflection high energy electron diffraction. See RHEED region fcc-like, 114 hcp-like, 114 relaxation adsorbate-induced, 436 step-induced, 436 reptation, 517, 539, 640, 652, 687 repulsive distance, 51 ring, 501 resonant enhanced multi-photon ionization, 169 RGL, 197, 207, 228, 235, 263, 326, 350, 354, 600, 630, 686, 716 approximation, 228, 236 Rh(100), 117, 153, 162, 305, 306, 307, 308, 423, 426, 539, 547, 577, 581, 593, 595, 596, 652 Rh(110), 87, 94, 98, 168, 203, 204, 596, 699 Rh(111), 304, 305, 306, 307, 449, 596 Rh(311), 87, 202, 426 Rh(331), 87 RHEED, 271, 323, 439, 451 rolling-over motion, 478 rotation atom-by-atom, 582 row shearing at the edges, 562 Ru(0001), 112, 304, 542, 678 Rutherford backscattering, 290 spectrometry, 198 saddle point, 20, 89, 113, 272, 546, 579, 622, 635, 650, 718 saturation island density, 110, 163, 184, 207, 268, 317 scanning electron microscopy, 315, 322 scanning tunneling microscopy, 24, 31, 32, 33, 34, 39, 46, 47, 48, 49, 53, 64, 75, 110, 112, 158, 162, 163, 166, 167, 168, 183, 188, 202, 205, 207, 209, 232, 235, 265, 271, 296, 301, 304, 312, 317, 321, 322, 323, 328, 350, 355, 356, 357, 363, 368, 372, 425, 428, 439, 440, 442, 450, 451, 455, 469, 478, 504, 575, 590, 598, 649, 667, 676, 688, 715, 720, 721, 722, 723, 735 atom tracking, 47 fast, 296, 637 image, 127, 163, 169, 528 low-temperature, 167 variable temparature, 317

755

Index

scattering elastic, 47 inelastic, 47 quasielastic, 207 Schlömilch relation, 14 screw dislocation, 506, 507 second-range transition, 138, 141 shearing, 538, 562, 687 motion, 626, 627, 628 shuffle gliding, 692 Si(100)–(2 × 1), 169 single jump frequency prefactor, 141 prefactor, 107 single transition, 119, 142, 210, 233 sintering, 696 site adsorption, 39, 54, 69, 71, 72, 73, 74, 75, 78, 80, 81, 129, 151, 261, 265, 277, 296, 300, 307, 314, 331, 347, 348, 350, 370, 477, 561, 601, 631, 643, 647, 711, 717 binding, 54, 68, 69, 72, 77, 81, 84, 105, 113, 114, 117, 152, 156, 262, 265, 275, 304, 344, 347, 357, 358, 598, 610, 636, 702, 704, 718 bridge, 68, 111, 573, 630, 686 deep, 225, 226 equilibrium, 17, 18, 46, 86, 98, 111, 537 fault, 68, 148, 261, 344 fcc, 22, 72, 73, 74, 75, 77, 78, 79, 80, 111, 115, 152, 166, 261, 265, 276, 277, 296, 299, 300, 301, 304, 306, 307, 313, 314, 317, 318, 319, 320, 321, 347, 348, 350, 351, 352, 353, 355, 357, 360, 370, 371, 428, 444, 472, 476, 477, 539, 564, 565, 566, 573, 575, 582, 585, 586, 599, 601, 623, 626, 636, 639, 640, 641, 642, 643, 644, 687, 717 four fold, 69, 80, 477, 632, 711 hcp, 22, 72, 73, 74, 75, 78, 80, 112, 115, 152, 261, 265, 277, 298, 299, 300, 301, 306, 307, 314, 317, 318, 319, 320, 321, 347, 348, 350, 352, 353, 355, 357, 360, 370, 371, 428, 444, 472, 473, 477, 539, 564, 565, 566, 573, 586, 587, 589, 598, 601, 623, 633, 636, 639, 640, 641, 643, 687 lattice, 68, 98, 148, 344, 345, 610, 611, 619 metastable, 477 normal, 5, 56, 98 shallow, 69, 226 size distribution, 49, 164, 311 sky-hook mechanism, 426, 428 smart Monte Carlo simulation, 287, 306 SPA-LEED. See LEED spot profile sphere-like shape, 651, 692 sputtering, 28, 33, 35, 96, 187, 210 stable location, 72 static potential barrier, 56 step A-type, 435, 437, 442, 446, 449, 453, 455, 456, 458, 460, 463, 464, 465, 467, 475, 476, 477, 478, 487, 501, 679

step (cont.) B-type, 434, 435, 437, 440, 441, 442, 445, 446, 449, 453, 458, 463, 465, 467, 473, 475, 476, 477, 478, 487, 500, 536, 679, 682 step flow, 442, 451, 467 step vacancy, 465 step-edge trap, 457, 479, 480, 481 STM. See scanning tunneling microscopy stochastic trajectory calculations, 148 strain, 86, 105, 111, 112, 113, 114, 115, 269, 272, 273, 281, 286, 295, 299, 316, 317, 323, 360, 450, 451, 453, 455, 509, 668, 727 biaxial, 115 compressive, 110, 111, 113, 295, 509, 564 local, 114, 115 surface, 111, 113 tensile, 111, 113, 115, 509 strain-induced event, 543 exchange, 326 stress, 33, 112, 299, 501, 590, 741 compressive, 453 tensile, 104, 112, 453 stress-induced mechanism, 648 stretching dimer, 546, 562, 595, 647 sub-lattice c(2 × 2), 103 sublimation energy, 57, 263 superposition technique, 68 surface defect, 201, 209, 217, 269, 314, 315, 332 surface ionization microscope, 223 surface-embedded atom method, 108, 600, 639 surface-state band, 720 Sutton-Chen model, 57 Ta(100), 328 Ta(110), 328, 404, 601, 652 Ta(111), 328 terrace width, 442 tetramer rotation, 623, 626, 633 translation, 623, 626, 633 thermal bath, 127 desorption, 149 energy, 135 equilibrium, 146 thermal energy He atom scattering, 283, 295, 316 thermal energy helium atom diffraction, 162 thermally assisted tunneling, 79 associated tunneling, 79 equilibrated surfaces, 110 equilibrated system, 162 third moment, 13 three-body effect, 713 attractive, 704

756

Index

tight-binding theory, 57 fourth-moment approximation, 218, 226, 331, 615, 711 RGL model, 325 second-moment approximation, 57, 195, 263, 268, 276, 300, 358, 370, 371, 372, 443, 448, 571, 673 TiO2(110)., 168 transient cooling, 604 heating, 604 hopping, 147 mobility, 155, 156, 159, 162, 163, 164, 166, 168, 169, 171 period, 44 state, 539 temperature, 42, 43, 44, 131, 134, 139, 221 time, 42 transition state, 17, 18, 58, 89, 104, 111, 112, 477, 544, 573, 574, 645, 687 barrier, 288 calculations, 276, 289 estimates, 286 theory, 4, 17, 122, 235, 236, 263, 266, 275, 279, 288, 293, 305, 309, 311, 317, 327, 331, 364, 366, 369, 595 translation, 19 translation with rotation, 651 translational motion, 573, 582, 596, 642 trimer bent form, 710, 713 linear form, 611, 626, 633, 643, 713 rotation, 626, 633, 640, 651 translation, 565, 566, 626, 639, 640, 643, 651 triangular form, 566, 569, 573, 611, 626, 633, 639, 643, 710, 713 trimer diffusivity prefactor, 643 trio-energy, 715 two-dimensional arrangement, 619 cluster, 483, 556, 613, 652 distribution of separations, 705 electron gas, 722 form, 593 island, 323, 468, 542, 569, 593, 602, 634, 653, 712 motion, 21, 88 movement, 93, 224, 530, 556 nucleation, 442, 451, 467 pentamer, 557 plane, 39 potential, 546 process, 22 shape, 613 spreading, 290 structure, 619, 652 surface, 100, 123, 183, 261, 596, 609

two-dimensional (cont.) system, 99 trimer, 622, 627 uncorrelated motion, 567 upstanding molecule, 168 upward exchange, 449 vacancy, 102, 210, 211, 324, 465, 469, 620, 671, 678 cluster, 553, 677, 678 island, 296, 451, 469, 478, 575, 598, 670, 671, 673, 676, 677, 689 worm-like cluster, 673 van der Waals forces, 148 VASP, 58, 200, 234, 272, 289, 301, 310, 496, 561, 563, 571, 580, 595, 598, 600, 634, 639, 646, 647, 648, 718, 728 code, 264 VC approximation, 184, 210, 233, 275, 279, 280, 286, 288 parameters, 288, 325 parametrization, 77 vertical dimer, 549 intermediate, 530, 531 mobility, 601, 648 states, 549 transition, 129, 131, 132, 133, 340 vibrational coupling, 696 vibrational frequency, 87, 685 effective, 215 Vienna ab initio package. See VASP Vineyard, 20, 686 expression, 685, 687 Vycor, 27, 35, 36 W(100), 272, 341, 342, 344 W(110), 31, 41, 45, 72, 83, 129, 130, 133, 138, 140, 141, 142, 145, 263, 315, 316, 319, 330, 331, 332, 333, 334, 335, 336, 337, 339, 340, 341, 347, 402, 403, 404, 423, 431, 432, 456, 502, 504, 507, 508, 557, 558, 608, 609, 611, 612, 613, 615, 617, 652, 664, 701, 703, 704, 706, 708, 711, 712, 715, 722, 726 W(111), 25, 68, 73, 148, 344, 345, 347, 456, 614 W(211), 38, 41, 45, 68, 81, 82, 87, 125, 126, 129, 141, 142, 145, 211, 212, 215, 216, 217, 218, 219, 221, 223, 224, 225, 226, 263, 402, 403, 433, 456, 525, 527, 558, 602, 603, 604, 606, 607, 610, 652, 699, 700, 710, 724 W(321), 68, 80, 87, 225, 226, 227, 429, 431, 603, 699 waiting time, 613 prefactor, 613 wave function, 696, 720 wedding cake structure, 429, 451 WIEN97, 294 work function, 28, 30, 53, 201, 704 change, 24, 200, 224, 278, 302, 332, 335, 338 Wulff polyhedron, 319, 326, 367

757

Index

XPD, 328 XPS, 328 “zero-time” distribution, 44 experiment, 43, 44, 45, 133, 141

“zero-time” (cont.) measurement, 44, 45, 139 results, 141 zigzag motion, 564, 599 step, 453 transition, 564